%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: modart03.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips modart03.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.11.26:1011 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 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Fx(The)f(language)e(of)g(the)h(logic)e(should)i(b)s(e)g(strong)f (\(in)g(the)h(sense)i(of)d(b)s(eing)g(expressiv)m(e\),)352 5855 y(but)33 b(the)g(existence)h(axioms)e(used)h(should)g(b)s(e)g(w)m (eak.)261 5971 y Fr(\017)41 b Fx(T)m(yp)s(e)h(parameters)f(\(ML)f(st)m (yle\))h(should)f(b)s(e)h(allo)m(w)m(ed,)h(but)e(quan)m(ti\014cation)g (o)m(v)m(er)h(t)m(yp)s(es)352 6087 y(should)25 b(b)s(e)f(disallo)m(w)m (ed)g(in)f(order)i(to)f(k)m(eep)i(the)f(theory)g(predicativ)m(e.)41 b(Similarly)-8 b(,)22 b(predicate)352 6204 y(v)-5 b(ariables)40 b(should)g(b)s(e)g(allo)m(w)m(ed)g(as)h(place-holders)f(for)f(prop)s (erties,)k(but)e(quan)m(ti\014cation)352 6320 y(o)m(v)m(er)34 b(them)e(should)h(b)s(e)g(disallo)m(w)m(ed,)e(again)g(to)i(ensure)h (predicativit)m(y)-8 b(.)1861 6419 y Fq(1)p eop %%Page: 2 2 2 1 bop 137 450 a Fx(Compared)24 b(with)g(the)g(literature,)h(the)f(no) m(v)m(el)h(asp)s(ect)f(of)g(the)g(presen)m(t)i(w)m(ork)f(is)f(the)g (dev)m(elopmen)m(t)38 566 y(of)h(elemen)m(tary)h(constructiv)m(e)i (analysis)d(in)g(suc)m(h)i(a)f(w)m(a)m(y)h(that)f(witnesses)i(ha)m(v)m (e)f(as)f(lo)m(w)g(a)f(t)m(yp)s(e)i(lev)m(el)38 683 y(as)22 b(p)s(ossible.)40 b(This)22 b(clearly)f(is)h(imp)s(ortan)m(t)f(for)g (the)i(complexit)m(y)f(of)g(the)g(extracted)i(programs.)39 b(Here)38 799 y(are)32 b(some)h(examples.)186 951 y(\(1\))41 b(A)33 b(con)m(tin)m(uous)f(function)g(on)g(the)g(reals)g(is)g (determined)g(b)m(y)h(its)e(v)-5 b(alues)32 b(on)g(the)h(rationals,)352 1067 y(and)g(hence)h(can)f(b)s(e)g(represen)m(ted)i(b)m(y)e(a)f(t)m(yp) s(e-one)i(\(rather)e(that)h(t)m(yp)s(e-t)m(w)m(o\))g(ob)5 b(ject.)186 1183 y(\(2\))41 b(In)33 b(the)f(pro)s(of)f(that)h(the)g (range)g(of)f(a)h(con)m(tin)m(uous)g(function)g(on)g(a)f(compact)h(in)m (terv)-5 b(all)29 b(has)352 1300 y(a)38 b(suprem)m(um)h(\(see)g (Corollary)e(3.11\),)i Fs(Br)n(ouwer)p Fx('s)g(notion)e(of)h(a)g Fw(total)5 b(ly)41 b(b)-5 b(ounde)g(d)48 b Fx(set)352 1416 y(of)38 b(reals)g(\(whic)m(h)h(has)g(t)m(yp)s(e-lev)m(el)g(t)m(w)m (o\))g(is)f(replaced)g(b)m(y)i(the)f(notion)e(of)h(b)s(eing)g Fw(b)-5 b(ounde)g(d)352 1532 y(ab)g(ove)39 b Fx(\(whic)m(h)33 b(has)g(t)m(yp)s(e-lev)m(el)g(one\).)186 1648 y(\(3\))41 b(The)36 b Fs(Ca)n(uchy)r Fx(-)p Fs(Euler)e Fx(construction)h(of)f (appro)m(ximate)g(solutions)g(to)g(ordinary)g(di\013er-)352 1764 y(en)m(tial)e(equations)g(can)h(b)s(e)g(seen)h(as)f(a)f(t)m(yp)s (e-lev)m(el)h(one)g(pro)s(cess.)38 2001 y Fp(Ac)m(kno)m(wledgemen)m (ts.)48 b Fx(Some)37 b(of)h(the)g(material)d(on)j(calculus)f(from)g(a)h (constructiv)m(e)h(p)s(oin)m(t)e(of)38 2117 y(view)46 b(is)e(tak)m(en)j(form)d([16].)82 b(The)46 b(section)f(on)h(ordinary)e (di\013eren)m(tial)g(equations)i(is)f(based)h(on)38 2233 y(Chapter)38 b(1)g(of)44 b Fs(Hurewicz)p Fx('s)38 b(textb)s(o)s(ok)g ([12)o(],)i(adapted)e(to)f(our)g(constructiv)m(e)i(setting.)59 b(I)37 b(ha)m(v)m(e)38 2349 y(also)31 b(made)i(use)g(of)f(a)h(recen)m (t)g(note)g(of)39 b Fs(Bridges)32 b 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b(or)40 b(every)i(r)-5 b(e)g(al)41 b Ft(x)f Fx(:=)g(\(\()p Ft(a)1673 5695 y Fu(n)1720 5680 y Fx(\))1758 5695 y Fu(n)1805 5680 y Ft(;)17 b(M)10 b Fx(\))42 b Fw(we)f(c)-5 b(an)41 b(\014nd)g(an)g(upp)-5 b(er)42 b(b)-5 b(ound)41 b Fx(2)3329 5643 y Fu(k)3366 5651 y Fk(x)3450 5680 y Fw(on)g(the)38 5796 y(elements)34 b(of)g(the)43 b Fs(Ca)n(uchy)35 b Fw(se)-5 b(quenc)g(e:)44 b Ft(a)1624 5811 y Fu(n)1699 5796 y Fr(\024)28 b Fx(2)1853 5760 y Fu(k)1890 5768 y Fk(x)1967 5796 y Fw(for)35 b(al)5 b(l)34 b Ft(n)p Fw(.)38 6000 y(Pr)-5 b(o)g(of.)41 b Fx(Let)g Ft(k)570 6015 y Fu(x)654 6000 y Fx(b)s(e)g(suc)m(h)i(that)d(max)p Fr(f)17 b Ft(a)1543 6015 y Fu(n)1631 6000 y Fr(j)41 b Ft(n)h Fr(\024)h Ft(M)10 b Fx(\(0\))17 b Fr(g)27 b Fx(+)h(1)41 b Fr(\024)h Fx(2)2605 5964 y Fu(k)2642 5972 y Fk(x)2685 6000 y Fx(,)h(hence)f Ft(a)3085 6015 y Fu(n)3174 6000 y Fr(\024)g Fx(2)3342 5964 y Fu(k)3379 5972 y Fk(x)3463 6000 y Fx(for)e(all)38 6116 y Ft(n)p Fx(.)p 3682 6057 41 4 v 3682 6113 4 56 v 3720 6113 V 3682 6116 41 4 v 137 6320 a(Clearly)32 b(this)g(assignmen)m(t)h(of)f Ft(k)1331 6335 y Fu(x)1407 6320 y Fx(to)g Ft(x)h Fx(is)f(not)h(compatible)d(with) i(equalit)m(y)-8 b(.)p eop %%Page: 3 3 3 2 bop 38 450 a Fp(De\014nition)36 b(1.5.)42 b Fx(A)32 b(real)g Ft(x)c Fx(:=)g(\(\()p Ft(a)1413 465 y Fu(n)1460 450 y Fx(\))1498 465 y Fu(n)1545 450 y Ft(;)17 b(M)10 b Fx(\))33 b(is)f(called)f Fw(nonne)-5 b(gative)39 b Fx(\(written)32 b Ft(x)c Fr(2)g Fn(R)3301 414 y Fm(0)q(+)3402 450 y Fx(\))k(if)1484 627 y Fr(8)p Ft(k)s Fx(\()p Fr(\000)q Fx(2)1758 586 y Fv(\000)p Fu(k)1883 627 y Fr(\024)c Ft(a)2039 643 y Fu(M)7 b Fm(\()p Fu(k)r Fm(\))2212 627 y Fx(\))p Ft(:)38 801 y Fx(It)32 b(is)h Ft(k)s Fx(-)p Fw(p)-5 b(ositive)38 b Fx(\(written)33 b Ft(x)28 b Fr(2)1213 816 y Fu(k)1284 801 y Fn(R)1350 765 y Fm(+)1415 801 y Fx(,)k(or)g Ft(x)d Fr(2)f Fn(R)43 b Fx(if)32 b Ft(k)j Fx(is)d(not)h(needed\))h(if)1570 978 y(2)1619 937 y Fv(\000)p Fu(k)1744 978 y Fr(\024)28 b Ft(a)1900 994 y Fu(M)7 b Fm(\()p Fu(k)r Fm(+1\))2163 978 y Ft(:)137 1155 y Fx(One)33 b(can)g(see)h(easily)e(that)g(b)s(oth)g Ft(x)d Fr(2)f Fn(R)1634 1118 y Fm(0+)1766 1155 y Fx(and)33 b Ft(x)28 b Fr(2)g Fn(R)2199 1118 y Fm(+)2297 1155 y Fx(are)k(compatible)f(with)h(equalit)m(y)-8 b(.)137 1271 y(Giv)m(en)36 b(real)f(n)m(um)m(b)s(ers)i Ft(x)c Fx(:=)g(\(\()p Ft(a)1364 1286 y Fu(n)1411 1271 y Fx(\))1449 1286 y Fu(n)1496 1271 y Ft(;)17 b(M)10 b Fx(\))36 b(and)g Ft(y)g Fx(:=)d(\(\()p Ft(b)2248 1286 y Fu(n)2295 1271 y Fx(\))2333 1286 y Fu(n)2380 1271 y Ft(;)17 b(N)10 b Fx(\),)37 b(w)m(e)g(de\014ne)g(eac)m(h)f Ft(z)41 b Fx(from)34 b(the)38 1387 y(list)c Ft(x)22 b Fx(+)f Ft(y)t Fx(,)31 b Fr(\000)p Ft(x)p Fx(,)i Fr(j)p Ft(x)p Fr(j)p Fx(,)f Ft(x)22 b Fr(\001)f Ft(y)t Fx(,)31 b(and)1284 1348 y Fm(1)p 1281 1364 40 4 v 1281 1421 a Fu(x)1363 1387 y Fx(\(the)h(latter)g(only)f(pro)m(vided)i(that)e Fr(j)p Ft(x)p Fr(j)d(2)2863 1402 y Fu(l)2917 1387 y Fn(R)2983 1351 y Fm(+)3048 1387 y Fx(\))k(as)g(represen)m(ted)38 1504 y(b)m(y)h(the)g(resp)s(ectiv)m(e)h(sequence)i(\()p Ft(c)1279 1519 y Fu(n)1326 1504 y Fx(\))c(of)g(rationals)f(with)h(mo)s (dulus)f Ft(L)p Fx(:)436 1676 y Ft(z)p 1107 1711 4 117 v 669 w(c)1192 1691 y Fu(n)p 1845 1711 V 1888 1676 a Ft(L)p Fx(\()p Ft(k)s Fx(\))p 395 1714 2972 4 v 436 1799 a Ft(x)23 b Fx(+)f Ft(y)p 1107 1834 4 120 v 490 w(a)1201 1814 y Fu(n)1270 1799 y Fx(+)g Ft(b)1409 1814 y Fu(n)p 1845 1834 V 1888 1799 a Fx(max)2070 1718 y Fj(\000)2116 1799 y Ft(M)10 b Fx(\()p Ft(k)25 b Fx(+)d(1\))p Ft(;)17 b(N)10 b Fx(\()p Ft(k)26 b Fx(+)c(1\))2951 1718 y Fj(\001)436 1915 y Fr(\000)p Ft(x)p 1107 1950 4 117 v 582 w Fr(\000)q Ft(a)1279 1930 y Fu(n)p 1845 1950 V 1888 1915 a Ft(M)10 b Fx(\()p Ft(k)s Fx(\))436 2031 y Fr(j)p Ft(x)p Fr(j)p 1107 2066 V 603 w(j)p Ft(a)1229 2046 y Fu(n)1276 2031 y Fr(j)p 1845 2066 V 584 w Ft(M)g Fx(\()p Ft(k)s Fx(\))436 2151 y Ft(x)23 b Fr(\001)f Ft(y)p 1107 2186 4 120 v 538 w(a)1201 2166 y Fu(n)1270 2151 y Fr(\001)g Ft(b)1361 2166 y Fu(n)p 1845 2186 V 1888 2151 a Fx(max)2070 2070 y Fj(\000)2116 2151 y Ft(M)10 b Fx(\()p Ft(k)s Fx(+1+)p Ft(k)2564 2166 y Fv(j)p Fu(y)r Fv(j)2644 2151 y Fx(\))p Ft(;)17 b(N)10 b Fx(\()p Ft(k)s Fx(+1+)p Ft(k)3158 2166 y Fv(j)p Fu(x)p Fv(j)3241 2151 y Fx(\))3279 2070 y Fj(\001)448 2321 y Fm(1)p 446 2337 40 4 v 446 2394 a Fu(x)528 2360 y Fx(for)32 b Fr(j)p Ft(x)p Fr(j)c(2)882 2375 y Fu(l)936 2360 y Fn(R)1002 2324 y Fm(+)p 1107 2484 4 299 v 1150 2190 a Fj(\()1263 2253 y Fm(1)p 1240 2270 80 4 v 1240 2327 a Fu(a)1277 2335 y Fk(n)1428 2293 y Fx(if)j Ft(a)1568 2308 y Fu(n)1643 2293 y Fr(6)p Fx(=)d(0)1230 2433 y(0)149 b(if)31 b Ft(a)1568 2448 y Fu(n)1643 2433 y Fx(=)d(0)p 1845 2484 4 299 v 1888 2360 a Ft(M)10 b Fx(\(2\()p Ft(l)25 b Fx(+)d(1\))g(+)g Ft(k)s Fx(\))p Ft(;)38 2632 y Fx(where)34 b(2)369 2596 y Fu(k)406 2604 y Fk(x)481 2632 y Fx(is)e(the)h(upp)s(er)g (b)s(ound)g(pro)m(vided)g(b)m(y)g(Lemma)e(1.4.)137 2748 y(One)36 b(can)g(sho)m(w)h(easily)e(that)g(for)g(reals)g Ft(x;)17 b(y)39 b Fx(also)c Ft(x)24 b Fx(+)g Ft(y)t Fx(,)36 b Fr(\000)p Ft(x)p Fx(,)h Fr(j)p Ft(x)p Fr(j)p Fx(,)f Ft(x)25 b Fr(\001)f Ft(y)38 b Fx(and)e(\(pro)m(vided)g(that)38 2864 y Fr(j)p Ft(x)p Fr(j)30 b(2)245 2879 y Fu(l)302 2864 y Fn(R)368 2828 y Fm(+)433 2864 y Fx(\))k(also)g(1)p Ft(=x)g Fx(are)g(reals,)h(and)f(moreo)m(v)m(er)h(that)f(he)h(functions) f Ft(x)24 b Fx(+)f Ft(y)t Fx(,)34 b Fr(\000)p Ft(x)p Fx(,)i Fr(j)p Ft(x)p Fr(j)p Fx(,)f Ft(x)23 b Fr(\001)g Ft(y)38 b Fx(and)38 2981 y(\(pro)m(vided)33 b(that)f Fr(j)p Ft(x)p Fr(j)c(2)893 2996 y Fu(l)947 2981 y Fn(R)1013 2944 y Fm(+)1078 2981 y Fx(\))k(also)g(1)p Ft(=x)g Fx(are)h(compatible) d(with)j(equalit)m(y)-8 b(.)38 3125 y Fp(Lemma)37 b(1.6.)42 b Fw(F)-7 b(or)34 b(r)-5 b(e)g(als)34 b Ft(x;)17 b(y)t(;)g(z)714 3328 y(x)23 b Fx(+)f(\()p Ft(y)j Fx(+)d Ft(z)t Fx(\))29 b(=)e(\()p Ft(x)c Fx(+)f Ft(y)t Fx(\))f(+)h Ft(z)963 3473 y(x)g Fx(+)g(0)28 b(=)f Ft(x)803 3624 y(x)22 b Fx(+)g(\()p Fr(\000)p Ft(x)p Fx(\))29 b(=)e(0)960 3769 y Ft(x)c Fx(+)f Ft(y)31 b Fx(=)c Ft(y)e Fx(+)d Ft(x)2163 3284 y(x)g Fr(\001)g Fx(\()p Ft(y)j Fr(\001)d Ft(z)t Fx(\))28 b(=)g(\()p Ft(x)23 b Fr(\001)e Ft(y)t Fx(\))h Fr(\001)f Ft(z)2363 3430 y(x)h Fr(\001)g Fx(1)27 b(=)h Ft(x)1890 3638 y Fx(0)g Ft(<)f Fr(j)p Ft(x)p Fr(j)h(!)f Ft(x)c Fr(\001)2477 3570 y Fx(1)p 2473 3615 56 4 v 2473 3706 a Ft(x)2566 3638 y Fx(=)28 b(1)2360 3812 y Ft(x)23 b Fr(\001)f Ft(y)30 b Fx(=)e Ft(y)d Fr(\001)d Ft(x)1366 3963 y(x)h Fr(\001)f Fx(\()p Ft(y)j Fx(+)d Ft(z)t Fx(\))28 b(=)g Ft(x)22 b Fr(\001)g Ft(y)k Fx(+)c Ft(x)g Fr(\001)g Ft(z)38 4150 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Routine.)p 3682 4090 41 4 v 3682 4146 4 56 v 3720 4146 V 3682 4149 41 4 v 38 4335 a Fp(Lemma)c(1.7.)42 b Fw(F)-7 b(or)34 b(r)-5 b(e)g(als)34 b Ft(x;)17 b(y)38 b Fw(fr)-5 b(om)35 b Ft(x)22 b Fr(\001)g Ft(y)31 b Fx(=)c(1)35 b Fw(we)f(c)-5 b(an)35 b(infer)f Fx(0)28 b Ft(<)f Fr(j)p Ft(x)p Fr(j)p Fw(.)38 4521 y(Pr)-5 b(o)g(of.)41 b Fx(Pic)m(k)30 b Ft(k)i Fx(suc)m(h)f(that)e Fr(j)p Ft(b)1128 4536 y Fu(n)1175 4521 y Fr(j)e(\024)h Fx(2)1384 4485 y Fu(k)1456 4521 y Fx(for)h(all)e Ft(n)p Fx(.)42 b(Pic)m(k)30 b Ft(q)j Fx(suc)m(h)e(that)e Ft(q)i Fr(\024)d Ft(n)i Fx(implies)d(1)p Ft(=)p Fx(2)g Fr(\024)h Ft(a)3502 4536 y Fu(n)3564 4521 y Fr(\001)15 b Ft(b)3648 4536 y Fu(n)3696 4521 y Fx(.)38 4637 y(Then)34 b(for)e Ft(q)f Fr(\024)d Ft(n)p Fx(,)33 b(1)p Ft(=)p Fx(2)27 b Fr(\024)h(j)p Ft(a)1097 4652 y Fu(n)1144 4637 y Fr(j)22 b(\001)g Fx(2)1293 4601 y Fu(k)1335 4637 y Fx(,)33 b(and)f(hence)i(2)1904 4601 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2120 4637 y Fr(\024)28 b(j)p Ft(a)2304 4652 y Fu(n)2351 4637 y Fr(j)p Fx(.)p 3682 4578 V 3682 4634 4 56 v 3720 4634 V 3682 4637 41 4 v 38 4823 a Fp(Lemma)37 b(1.8.)42 b Fw(F)-7 b(or)34 b(r)-5 b(e)g(als)34 b Ft(x;)17 b(y)t Fw(,)38 4968 y Fx(\(a\))41 b Ft(x;)17 b(y)31 b Fr(2)d Fn(R)542 4932 y Fm(0+)670 4968 y Fr(!)f Ft(x)c Fx(+)f Ft(y)t(;)17 b(x)k Fr(\001)h Ft(y)31 b Fr(2)d Fn(R)1434 4932 y Fm(0+)1534 4968 y Fw(,)32 5084 y Fx(\(b\))42 b Ft(x;)17 b(y)31 b Fr(2)d Fn(R)542 5048 y Fm(+)635 5084 y Fr(!)f Ft(x)c Fx(+)f Ft(y)t(;)17 b(x)k Fr(\001)h Ft(y)31 b Fr(2)d Fn(R)1399 5048 y Fm(+)1464 5084 y Fw(,)43 5200 y Fx(\(c\))42 b Ft(x)28 b Fr(2)g Fn(R)447 5164 y Fm(0+)575 5200 y Fr(!)f(\000)p Ft(x)i Fr(2)f Fn(R)1023 5164 y Fm(0+)1151 5200 y Fr(!)f Ft(x)h Fx(=)g(0)p Fw(.)38 5386 y(Pr)-5 b(o)g(of.)41 b Fx(\(a\),)50 b(\(b\).)85 b(Routine.)f(\(c\).)i(Let)46 b Ft(k)k Fx(b)s(e)c(giv)m(en.)86 b(Pic)m(k)47 b Ft(p)f Fx(suc)m(h)i(that)e Fr(\000)p Fx(2)3143 5350 y Fv(\000)p Fu(k)3292 5386 y Fr(\024)52 b Ft(a)3472 5401 y Fu(n)3566 5386 y Fx(and)38 5503 y Fr(\000)p Fx(2)164 5466 y Fv(\000)p Fu(k)289 5503 y Fr(\024)28 b(\000)p Ft(a)522 5518 y Fu(n)603 5503 y Fx(for)k Ft(n)c Fr(\025)g Ft(p)p Fx(.)43 b(Then)34 b Fr(j)p Ft(a)1396 5518 y Fu(n)1443 5503 y Fr(j)27 b(\024)h Fx(2)1652 5466 y Fv(\000)p Fu(k)1750 5503 y Fx(.)p 3682 5443 V 3682 5499 4 56 v 3720 5499 V 3682 5502 41 4 v 137 5688 a(W)-8 b(e)30 b(write)f Ft(x)f Fr(\024)g Ft(y)k Fx(for)d Ft(y)18 b Fr(\000)d Ft(x)29 b Fr(2)f Fn(R)1364 5652 y Fm(0+)1494 5688 y Fx(and)h Ft(x)f(<)g(y)k Fx(for)c Ft(y)19 b Fr(\000)c Ft(x)29 b Fr(2)f Fn(R)2495 5652 y Fm(+)2560 5688 y Fx(.)42 b(Un)m(winding)29 b(the)g(de\014nitions)38 5805 y(yields)e(that)g Ft(x)h Fr(\024)g Ft(y)j Fx(is)c(to)g(sa)m(y)i (that)e(for)g(ev)m(ery)i Ft(k)i Fx(there)d(exists)g Ft(p)2392 5820 y Fu(k)2462 5805 y Fx(suc)m(h)h(that)e Ft(a)2934 5820 y Fu(n)3009 5805 y Fr(\024)h Ft(b)3155 5820 y Fu(n)3215 5805 y Fx(+)12 b(2)3352 5768 y Fv(\000)p Fu(k)3476 5805 y Fx(for)27 b(all)38 5921 y Ft(n)i Fr(\025)h Ft(p)281 5936 y Fu(k)324 5921 y Fx(.)46 b(F)-8 b(urthermore,)33 b Ft(x)d(<)f(y)37 b Fx(is)c(a)g(shorthand)h(for)f(the)h(presence)i(of)d Ft(k)s(;)17 b(q)37 b Fx(with)c Ft(a)3183 5936 y Fu(n)3253 5921 y Fx(+)23 b(2)3401 5885 y Fv(\000)p Fu(k)3528 5921 y Fr(\024)29 b Ft(b)3675 5936 y Fu(n)38 6037 y Fx(for)j(all)e Ft(n)e Fr(\025)g Ft(q)t Fx(;)k(w)m(e)i(then)f(write)f Ft(x)c(<)1393 6052 y Fu(k)r(;q)1516 6037 y Ft(y)36 b Fx(\(or)c(simply)f Ft(x)d(<)2228 6052 y Fu(k)2298 6037 y Ft(y)36 b Fx(if)31 b Ft(q)36 b Fx(is)c(not)g(needed\))i(whenev)m(er)h (w)m(e)38 6153 y(w)m(an)m(t)e(to)f(call)f(these)j(witnesses.)38 6298 y Fp(Lemma)j(1.9.)42 b Fx(0)27 b Fr(\024)h Ft(x)36 b Fw(and)e Fx(0)27 b Ft(<)1264 6313 y Fu(k)1335 6298 y Ft(y)38 b Fw(implies)33 b Fx(0)28 b Ft(<)1907 6313 y Fu(k)r Fm(+1)2067 6298 y Ft(x)23 b Fx(+)f Ft(y)t Fw(.)p eop %%Page: 4 4 4 3 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(F)-8 b(rom)31 b(0)d Fr(\024)h Ft(x)k Fx(w)m(e)h(ha)m(v)m(e)g Fr(8)p Ft(l)r Fr(9)p Ft(p)p Fr(8)p Ft(n:)c(p)e Fr(\024)g Ft(n)h Fr(!)e(\000)p Fx(2)2114 414 y Fv(\000)p Fu(l)2224 450 y Fr(\024)h Ft(a)2380 465 y Fu(n)2427 450 y Fx(.)45 b(F)-8 b(rom)31 b(0)d Ft(<)2908 465 y Fu(k)2978 450 y Ft(y)36 b Fx(w)m(e)e(ha)m(v)m(e)g(some)f Ft(q)38 566 y Fx(suc)m(h)c(that)g Fr(8)p Ft(n:)f(q)j Fr(\024)e Ft(n)e Fr(!)h Fx(2)1071 530 y Fv(\000)p Fu(k)1196 566 y Fr(\024)g Ft(b)1342 581 y Fu(n)1389 566 y Fx(.)42 b(Pic)m(k)29 b Ft(p)f Fx(for)g Ft(k)16 b Fx(+)d(1.)42 b(Then)30 b Ft(p;)17 b(q)31 b Fr(\024)d Ft(n)g Fx(implies)e(0)i Fr(\024)g Ft(a)3337 581 y Fu(n)3397 566 y Fx(+)13 b(2)3535 530 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)38 683 y Fx(and)32 b(2)276 646 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)492 683 y Fr(\024)c Ft(b)638 698 y Fu(n)707 683 y Fr(\000)23 b Fx(2)856 646 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1044 683 y Fx(,)32 b(hence)i(2)1423 646 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1638 683 y Fr(\024)28 b Ft(a)1794 698 y Fu(n)1864 683 y Fx(+)22 b Ft(b)2003 698 y Fu(n)2050 683 y Fx(.)p 3682 623 41 4 v 3682 679 4 56 v 3720 679 V 3682 682 41 4 v 38 860 a Fp(Lemma)37 b(1.10.)42 b Fw(F)-7 b(or)34 b(r)-5 b(e)g(als)34 b Ft(x;)17 b(y)t(;)g(z)t Fw(,)569 1006 y Ft(x)28 b Fr(\024)h Ft(x)569 1151 y(x)f Fr(\024)h Ft(y)h Fr(!)e Ft(y)j Fr(\024)d Ft(x)g Fr(!)f Ft(x)h Fx(=)g Ft(y)569 1297 y(x)g Fr(\024)h Ft(y)h Fr(!)e Ft(y)j Fr(\024)d Ft(z)k Fr(!)27 b Ft(x)i Fr(\024)f Ft(z)569 1442 y(x)g Fr(\024)h Ft(y)h Fr(!)e Ft(x)22 b Fx(+)g Ft(z)33 b Fr(\024)28 b Ft(y)d Fx(+)d Ft(z)569 1587 y(x)28 b Fr(\024)h Ft(y)h Fr(!)e Fx(0)f Fr(\024)h Ft(z)33 b Fr(!)27 b Ft(x)c Fr(\001)e Ft(z)33 b Fr(\024)28 b Ft(y)d Fr(\001)d Ft(z)1933 1079 y(x)28 b Fr(6)p Ft(<)f(x)1933 1224 y(x)h(<)f(y)k Fr(!)c Ft(y)k(<)d(z)k Fr(!)27 b Ft(x)i(<)e(z)1933 1369 y(x)h(<)f(y)k Fr(!)c Ft(x)c Fx(+)f Ft(z)32 b(<)c(y)d Fx(+)d Ft(z)1933 1515 y(x)28 b(<)f(y)k Fr(!)c Fx(0)h Ft(<)f(z)33 b Fr(!)27 b Ft(x)c Fr(\001)e Ft(z)33 b(<)27 b(y)f Fr(\001)21 b Ft(z)38 1765 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Lemma)31 b(1.8.)p 3682 1705 V 3682 1761 4 56 v 3720 1761 V 3682 1764 41 4 v 38 1942 a Fp(Lemma)37 b(1.11.)42 b Ft(x)28 b Fr(\024)g Ft(y)j Fr(!)c Ft(y)k(<)1256 1957 y Fu(k)1326 1942 y Ft(z)i Fr(!)27 b Ft(x)h(<)1690 1957 y Fu(k)r Fm(+1)1850 1942 y Ft(z)t Fw(.)38 2119 y(Pr)-5 b(o)g(of.)41 b Fx(Lemma)31 b(1.9.)p 3682 2060 V 3682 2115 4 56 v 3720 2115 V 3682 2118 41 4 v 137 2296 a(As)25 b(is)e(to)g(b)s(e)h(exp)s(ected)h(in)e (view)h(of)f(the)h(existen)m(tial)f(and)g(univ)m(ersal)h(c)m(haracter)g (of)f(the)h(predicates)38 2412 y Ft(<)32 b Fx(and)h Fr(\024)g Fx(on)f(the)h(reals,)g(w)m(e)g(ha)m(v)m(e:)38 2553 y Fp(Lemma)k(1.12.)42 b Ft(x)28 b Fr(\024)g Ft(y)j Fr($)c Ft(y)k Fr(6)p Ft(<)d(x)p Fw(.)38 2730 y(Pr)-5 b(o)g(of.)41 b Fr(!)p Fx(.)49 b(Assume)36 b Ft(x)c Fr(\024)f Ft(y)38 b Fx(and)d Ft(y)f(<)e(x)p Fx(.)50 b(By)35 b(Lemma)e(1.11)h(w)m(e)i (obtain)e Ft(x)e(<)f(x)p Fx(.)50 b(con)m(tradicting)38 2847 y(Lemma)31 b(1.10.)137 2963 y Fr( )p Fx(.)57 b(It)37 b(clearly)f(su\016ces)j(to)e(sho)m(w)h(0)d Fr(6)p Ft(<)g(z)40 b Fr(!)35 b Ft(z)40 b Fr(\024)35 b Fx(0,)j(for)f(a)f(real)h Ft(z)k Fx(giv)m(en)c(b)m(y)h(\()p Ft(c)3178 2978 y Fu(n)3225 2963 y Fx(\))3263 2978 y Fu(n)3310 2963 y Fx(.)57 b(Assume)38 3079 y(0)38 b Fr(6)p Ft(<)g(z)t Fx(.)63 b(W)-8 b(e)39 b(m)m(ust)g(sho)m(w)h Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(n:)f(p)g Fr(\024)f Ft(n)h Fr(!)f Ft(c)1920 3094 y Fu(n)2005 3079 y Fr(\024)h Fx(2)2170 3043 y Fv(\000)p Fu(k)2267 3079 y Fx(.)62 b(Let)39 b Ft(k)j Fx(b)s(e)d(giv)m(en.)62 b(By)39 b(assumption)38 3195 y(0)d Fr(6)p Ft(<)f(z)t Fx(,)40 b(hence)f Fr(:9)p Ft(l)r Fx(\(2)865 3159 y Fv(\000)p Fu(l)982 3195 y Fr(\024)e Ft(c)1138 3211 y Fu(M)7 b Fm(\()p Fu(l)q Fm(+1\))1384 3195 y Fx(\),)38 b(hence)h Fr(8)p Ft(l)r Fx(\()p Ft(c)1929 3211 y Fu(M)7 b Fm(\()p Fu(l)q Fm(+1\))2212 3195 y Ft(<)36 b Fx(2)2373 3159 y Fv(\000)p Fu(l)2453 3195 y Fx(\).)58 b(F)-8 b(or)37 b Ft(l)h Fx(:=)e Ft(k)29 b Fx(+)c(1)37 b(this)g(implies)38 3318 y Ft(c)80 3333 y Fu(M)7 b Fm(\()p Fu(k)r Fm(+2\))370 3318 y Ft(<)28 b Fx(2)523 3282 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)710 3318 y Fx(,)33 b(hence)h Ft(c)1083 3333 y Fu(n)1157 3318 y Fr(\024)28 b Ft(c)1304 3333 y Fu(M)7 b Fm(\()p Fu(k)r Fm(+2\))1589 3318 y Fx(+)22 b(2)1736 3282 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)1951 3318 y Ft(<)28 b Fx(2)2104 3282 y Fv(\000)p Fu(k)2234 3318 y Fx(for)k Ft(M)10 b Fx(\()p Ft(k)26 b Fx(+)c(2\))27 b Fr(\024)h Ft(n)p Fx(.)p 3682 3259 V 3682 3315 4 56 v 3720 3315 V 3682 3318 41 4 v 137 3495 a(Constructiv)m(ely)-8 b(,)35 b(w)m(e)f(cannot)g(compare)f (t)m(w)m(o)h(reals,)f(but)h(w)m(e)g(can)g(compare)e(ev)m(ery)k(real)c (with)h(a)38 3611 y(non)m(trivial)d(in)m(terv)-5 b(al.)38 3752 y Fp(Lemma)28 b(1.13)g Fx(\(Appro)m(ximate)c(Splitting)e (Principle\))p Fp(.)34 b Fw(L)-5 b(et)28 b Ft(x;)17 b(y)t(;)g(z)31 b Fw(b)-5 b(e)27 b(given)g(and)f(assume)h Ft(x)h(<)g(y)t Fw(.)38 3869 y(Then)34 b(we)g(c)-5 b(an)35 b(\014nd)f Ft(k)s(;)17 b(q)38 b Fw(such)d(that)g(either)g Ft(z)d(<)1841 3884 y Fu(k)r(;q)1965 3869 y Ft(y)38 b Fw(or)d Ft(x)28 b(<)2336 3884 y Fu(k)r(;q)2460 3869 y Ft(z)t Fw(.)38 4046 y(Pr)-5 b(o)g(of.)41 b Fx(Let)30 b Ft(x)e Fx(:=)g(\(\()p Ft(a)849 4061 y Fu(n)896 4046 y Fx(\))934 4061 y Fu(n)981 4046 y Ft(;)17 b(M)10 b Fx(\),)31 b Ft(y)g Fx(:=)d(\(\()p Ft(b)1552 4061 y Fu(n)1599 4046 y Fx(\))1637 4061 y Fu(n)1684 4046 y Ft(;)17 b(N)10 b Fx(\),)31 b Ft(z)h Fx(:=)c(\(\()p Ft(c)2238 4061 y Fu(n)2284 4046 y Fx(\))2322 4061 y Fu(n)2369 4046 y Ft(;)17 b(L)p Fx(\).)43 b(Assume)31 b Ft(p;)17 b(k)33 b Fx(are)d(giv)m(en)g(suc)m(h)38 4162 y(that)i(with)g Ft(")c Fx(:=)f(2)724 4126 y Fv(\000)p Fu(k)1210 4287 y Fr(8)p Ft(n:)h(p)g Fr(\024)g Ft(n)g Fr(!)f Ft(a)1824 4302 y Fu(n)1894 4287 y Fx(+)22 b(3)p Ft(")27 b Fr(\024)h Ft(b)2260 4302 y Fu(n)2329 4287 y Fr(\000)23 b Fx(3)p Ft(":)38 4433 y Fx(Let)33 b Ft(q)e Fx(:=)d(max)599 4352 y Fj(\000)645 4433 y Ft(M)10 b Fx(\()p Ft(k)s Fx(\))p Ft(;)17 b(N)10 b Fx(\()p Ft(k)s Fx(\))p Ft(;)17 b(L)p Fx(\()p Ft(k)s Fx(\))p Ft(;)g(p)1474 4352 y Fj(\001)1520 4433 y Fx(.)137 4549 y Fw(Case)40 b Ft(c)424 4564 y Fu(q)489 4549 y Fr(\024)28 b Ft(b)635 4564 y Fu(q)696 4549 y Fr(\000)23 b Fx(3)p Ft(")p Fx(.)43 b(Then)33 b(for)f Ft(q)g Fr(\024)c Ft(n)1225 4715 y(c)1267 4730 y Fu(n)1336 4715 y Fx(+)22 b Ft(")28 b Fr(\024)g Ft(c)1655 4730 y Fu(q)1715 4715 y Fx(+)22 b(2)p Ft(")27 b Fr(\024)h Ft(b)2081 4730 y Fu(q)2142 4715 y Fr(\000)22 b Ft(")28 b Fr(\024)g Ft(b)2461 4730 y Fu(n)2508 4715 y Ft(;)38 4881 y Fx(hence)34 b Ft(z)e(<)462 4896 y Fu(k)r(;q)586 4881 y Ft(y)t Fx(.)137 4997 y Fw(Case)40 b Ft(b)423 5012 y Fu(q)483 4997 y Fr(\000)23 b Fx(3)p Ft(")k(<)h(c)851 5012 y Fu(q)889 4997 y Fx(,)k(hence)i(b)m(y)g (assumption)e Ft(a)1922 5012 y Fu(q)1982 4997 y Fx(+)22 b(3)p Ft(")27 b(<)h(c)2348 5012 y Fu(q)2386 4997 y Fx(.)43 b(Then)34 b(for)e Ft(q)f Fr(\024)e Ft(n)1216 5163 y(a)1267 5178 y Fu(n)1337 5163 y Fx(+)22 b Ft(")27 b Fr(\024)h Ft(a)1664 5178 y Fu(q)1724 5163 y Fx(+)22 b(2)p Ft(")28 b(<)f(c)2090 5178 y Fu(q)2150 5163 y Fr(\000)c Ft(")k Fr(\024)h Ft(c)2470 5178 y Fu(n)2517 5163 y Ft(;)38 5329 y Fx(hence)34 b Ft(x)28 b(<)468 5344 y Fu(k)r(;q)592 5329 y Ft(z)t Fx(.)p 3682 5270 V 3682 5326 4 56 v 3720 5326 V 3682 5329 41 4 v 137 5506 a(Notice)34 b(that)g(the)g(b)s(o)s (olean)e(ob)5 b(ject)35 b(determining)d(whether)j Ft(z)g(<)30 b(y)37 b Fx(or)c Ft(x)e(<)e(z)39 b Fx(dep)s(ends)d(on)d(the)38 5622 y(represen)m(tation)42 b(of)g Ft(x)g Fx(and)g Ft(y)t Fx(.)71 b(In)42 b(particular)e(this)i(assignmen)m(t)f(is)h(not)f (compatible)f(with)i(our)38 5739 y(equalit)m(y)32 b(relation.)137 5855 y(One)23 b(migh)m(t)e(think)h(that)g(the)h(non-a)m(v)-5 b(ailable)20 b(comparison)h(of)h(t)m(w)m(o)h(reals)f(could)g(b)s(e)g (circum)m(v)m(en)m(ted)38 5971 y(b)m(y)i(using)f(a)h(maxim)m(um)d (function.)40 b(Indeed,)27 b(suc)m(h)e(a)e(function)g(can)g(easily)g(b) s(e)h(de\014ned)h(\(comp)s(onen)m(t)38 6087 y(wise\),)33 b(and)f(it)g(has)h(the)g(exp)s(ected)h(prop)s(erties)f Ft(x;)17 b(y)31 b Fr(\024)d Fx(max\()p Ft(x;)17 b(y)t Fx(\))31 b(and)i Ft(x;)17 b(y)31 b Fr(\024)d Ft(z)33 b Fr(!)27 b Fx(max\()p Ft(x;)17 b(y)t Fx(\))26 b Fr(\024)38 6204 y Ft(z)t Fx(.)41 b(Ho)m(w)m(ev)m(er,)27 b(what)c(is)g(missing)e (is)i(the)g(kno)m(wledge)h(that)f(max\()p Ft(x;)17 b(y)t Fx(\))22 b(equals)h(one)g(of)g(its)g(argumen)m(ts,)38 6320 y(i.e.,)32 b(w)m(e)i(do)e(not)h(ha)m(v)m(e)g(max\()p Ft(x;)17 b(y)t Fx(\))27 b(=)g Ft(x)c Fr(_)g Fx(max)o(\()p Ft(x;)17 b(y)t Fx(\))27 b(=)g Ft(y)t Fx(.)p eop %%Page: 5 5 5 4 bop 137 450 a Fx(Ho)m(w)m(ev)m(er,)37 b(in)c(man)m(y)g(cases)j(it)c (is)i(su\016cien)m(t)g(to)g(pic)m(k)g(the)g(up)g(to)f Ft(")h Fx(largest)f(real)g(out)g(of)g(\014nitely)38 566 y(man)m(y)38 b(giv)m(en)h(ones.)61 b(This)39 b(is)f(indeed)g(p)s (ossible.)60 b(W)-8 b(e)39 b(giv)m(e)g(the)f(pro)s(of)g(for)g(t)m(w)m (o)h(reals;)i(it)c(can)i(b)s(e)38 683 y(easily)32 b(generalized.)38 823 y Fp(Lemma)43 b(1.14)g Fx(\(Maxim)m(um)36 b(of)h(Tw)m(o)h(Reals\))p Fp(.)44 b Fw(L)-5 b(et)39 b Ft(x)e Fx(:=)f(\(\()p Ft(a)2402 838 y Fu(n)2449 823 y Fx(\))2487 838 y Fu(n)2534 823 y Ft(;)17 b(M)10 b Fx(\))40 b Fw(and)e Ft(y)h Fx(:=)d(\(\()p Ft(b)3296 838 y Fu(n)3344 823 y Fx(\))3382 838 y Fu(n)3429 823 y Ft(;)17 b(N)10 b Fx(\))39 b Fw(b)-5 b(e)38 939 y(r)g(e)g(als,)34 b(and)g Ft(k)d Fr(2)d Fn(N)9 b Fw(.)51 b(Then)34 b(for)h Ft(")27 b Fx(:=)h(2)1468 903 y Fv(\000)p Fu(k)1600 939 y Fw(either)34 b Ft(x)29 b Fr(\024)f Ft(y)d Fx(+)d Ft(")35 b Fw(or)f(else)h Ft(y)30 b Fr(\024)f Ft(x)22 b Fx(+)g Ft(")p Fw(.)38 1115 y(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(p)27 b Fx(:=)h(max)900 1034 y Fj(\000)945 1115 y Ft(M)10 b Fx(\()p Ft(k)26 b Fx(+)c(1\))p Ft(;)17 b(N)10 b Fx(\()p Ft(k)25 b Fx(+)d(1\))1780 1034 y Fj(\001)1826 1115 y Fx(.)137 1231 y Fw(Case)40 b Ft(a)433 1246 y Fu(p)500 1231 y Fr(\024)28 b Ft(b)646 1246 y Fu(p)686 1231 y Fx(.)44 b(Then)34 b(for)e Ft(p)27 b Fr(\024)h Ft(n)1216 1426 y(a)1267 1441 y Fu(n)1342 1426 y Fr(\024)g Ft(a)1498 1441 y Fu(p)1560 1426 y Fx(+)1670 1358 y Ft(")p 1668 1403 49 4 v 1668 1494 a Fx(2)1755 1426 y Fr(\024)g Ft(b)1901 1441 y Fu(p)1963 1426 y Fx(+)2073 1358 y Ft(")p 2071 1403 V 2071 1494 a Fx(2)2158 1426 y Fr(\024)g Ft(b)2304 1441 y Fu(n)2373 1426 y Fx(+)22 b Ft(":)38 1620 y Fx(This)33 b(holds)f(for)g(all)e Ft(n)p Fx(,)j(therefore)g Ft(x)28 b Fr(\024)h Ft(y)c Fx(+)d Ft(")p Fx(.)137 1736 y Fw(Case)40 b Ft(b)423 1751 y Fu(p)490 1736 y Ft(<)28 b(a)645 1751 y Fu(p)685 1736 y Fx(.)43 b(Then)34 b(for)e Ft(p)c Fr(\024)g Ft(n)1217 1931 y(b)1258 1946 y Fu(n)1333 1931 y Fr(\024)h Ft(b)1480 1946 y Fu(p)1542 1931 y Fx(+)1651 1864 y Ft(")p 1650 1908 V 1650 2000 a Fx(2)1736 1931 y Ft(<)f(a)1891 1946 y Fu(p)1953 1931 y Fx(+)2062 1864 y Ft(")p 2061 1908 V 2061 2000 a Fx(2)2147 1931 y Fr(\024)g Ft(a)2303 1946 y Fu(n)2373 1931 y Fx(+)22 b Ft(":)38 2126 y Fx(This)33 b(holds)f(for)g(all)e Ft(n)p Fx(,)j(therefore)g Ft(y)e Fr(\024)d Ft(x)23 b Fx(+)f Ft(")p Fx(.)p 3682 2066 41 4 v 3682 2122 4 56 v 3720 2122 V 3682 2125 41 4 v 137 2301 a(Recall)46 b(that)i(ev)m(ery)h(rational)c Ft(a)j Fx(is)f(tacitly)f(understo)s(o)s(d)i(as)g(the)g(real)f(represen)m(ted)j (b)m(y)e(the)38 2417 y(constan)m(t)33 b(sequence)j Ft(a)887 2432 y Fu(n)961 2417 y Fx(=)28 b Ft(a)33 b Fx(with)f(the)h(constan)m(t) g(mo)s(dulus)f Ft(M)10 b Fx(\()p Ft(k)s Fx(\))28 b(=)g(0.)38 2558 y Fp(Lemma)k(1.15)h Fx(\()p Fn(Q)46 b Fx(is)27 b(dense)j(in)e Fn(R)5 b Fx(\))p Fp(.)44 b Fw(F)-7 b(or)30 b(any)h(two)g(r)-5 b(e)g(als)31 b Ft(x)d(<)f(y)34 b Fw(ther)-5 b(e)31 b(is)g(a)g(r)-5 b(ational)30 b Ft(a)i Fw(such)e(that)38 2674 y Ft(x)e(<)f(a)h(<)g(y)t Fw(.)38 2849 y(Pr)-5 b(o)g(of.)41 b Fx(Let)29 b Ft(z)k Fx(:=)27 b(\()p Ft(x)15 b Fx(+)g Ft(y)t Fx(\))p Ft(=)p Fx(2)29 b(b)s(e)g(giv)m(en)g(b)m(y)h(\()p Ft(c)1723 2864 y Fu(n)1770 2849 y Fx(\))1808 2864 y Fu(n)1855 2849 y Fx(.)42 b(Then)31 b(for)d(some)h Ft(k)k Fx(w)m(e)d(ha)m(v)m(e)g Ft(x)e(<)3166 2864 y Fu(k)3237 2849 y Ft(z)k(<)3390 2864 y Fu(k)3460 2849 y Ft(y)t Fx(.)42 b(Let)38 2966 y Ft(a)28 b Fx(:=)f Ft(c)289 2981 y Fu(M)7 b Fm(\()p Fu(k)r Fm(+1\))552 2966 y Fx(.)p 3682 2907 V 3682 2962 4 56 v 3720 2962 V 3682 2965 41 4 v 137 3141 a(Notice)33 b(that)f Ft(a)h Fx(dep)s(ends)h(on)e(the)h(represen)m(tations)h(of)e Ft(x)h Fx(and)g Ft(y)t Fx(.)38 3282 y Fp(Theorem)k(1.16)h Fx(\()p Fs(Cantor)p Fx(\))p Fp(.)43 b Fw(L)-5 b(et)35 b(a)g(se)-5 b(quenc)g(e)34 b Fx(\()p Ft(x)2024 3297 y Fu(n)2072 3282 y Fx(\))h Fw(of)g(r)-5 b(e)g(als)34 b(b)-5 b(e)35 b(given.)45 b(Then)34 b(we)h(c)-5 b(an)34 b(\014nd)h(a)38 3398 y(r)-5 b(e)g(al)34 b Ft(y)k Fw(with)d Fx(0)28 b Fr(\024)g Ft(y)i Fr(\024)f Fx(1)34 b Fw(that)i(is)e(ap)-5 b(art)35 b(fr)-5 b(om)34 b(every)h Ft(x)2070 3413 y Fu(n)2118 3398 y Fw(,)f(in)h(the)g(sense)f(that)h Ft(x)2977 3413 y Fu(n)3052 3398 y Ft(<)28 b(y)d Fr(_)d Ft(y)31 b(<)d(x)3555 3413 y Fu(n)3602 3398 y Fw(.)38 3573 y(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)33 b(construct)h(t)m(w)m(o)f(sequences)j(\()p Ft(a)1649 3588 y Fu(n)1696 3573 y Fx(\))1734 3588 y Fu(n)1781 3573 y Fx(,)c(\()p Ft(b)1919 3588 y Fu(n)1967 3573 y Fx(\))2005 3588 y Fu(n)2084 3573 y Fx(of)g(rationals)f(suc)m(h)j(that)e (for)g(all)f Ft(n)842 3738 y Fx(0)c(=)h Ft(a)1073 3753 y Fm(0)1140 3738 y Fr(\024)g Ft(a)1296 3753 y Fm(1)1363 3738 y Fr(\024)h(\001)17 b(\001)g(\001)25 b(\024)k Ft(a)1769 3753 y Fu(n)1843 3738 y Ft(<)f(b)1988 3753 y Fu(n)2063 3738 y Fr(\024)g(\001)17 b(\001)g(\001)26 b(\024)i Ft(b)2458 3753 y Fm(1)2526 3738 y Fr(\024)g Ft(b)2672 3753 y Fm(0)2740 3738 y Fx(=)f(1)p Ft(;)-2881 b Fx(\(1\))842 3889 y Ft(x)897 3904 y Fu(n)972 3889 y Ft(<)27 b(a)1126 3904 y Fu(n)p Fm(+1)1286 3889 y Fr(_)22 b Ft(b)1415 3904 y Fu(n)p Fm(+1)1581 3889 y Ft(<)27 b(x)1739 3904 y Fu(n)1786 3889 y Ft(;)-1775 b Fx(\(2\))842 4047 y Ft(b)883 4062 y Fu(n)952 4047 y Fr(\000)23 b Ft(a)1103 4062 y Fu(n)1178 4047 y Fr(\024)28 b Fx(2)1332 4006 y Fv(\000)p Fu(n)1433 4047 y Ft(:)-1422 b Fx(\(3\))38 4217 y(Let)40 b Ft(a)271 4232 y Fm(0)311 4217 y Ft(;)17 b(:)g(:)g(:)f(;)h(a)581 4232 y Fu(n)668 4217 y Fx(and)40 b Ft(b)906 4232 y Fm(0)946 4217 y Ft(;)17 b(:)g(:)g(:)f(;)h(b)1206 4232 y Fu(n)1293 4217 y Fx(b)s(e)41 b(already)f(constructed)i(suc)m(h)g(that)e(\(1\))o(-\(3\))g(hold)f (\(as)i(far)f(as)38 4334 y(they)33 b(are)g(de\014ned\).)45 b(No)m(w)33 b(compare)f(the)h(real)f Ft(x)1857 4349 y Fu(n)1937 4334 y Fx(with)g Ft(a)2210 4349 y Fu(n)2285 4334 y Ft(<)27 b(b)2429 4349 y Fu(n)2477 4334 y Fx(.)137 4450 y Fw(Case)41 b Fx(1.)47 b Ft(x)561 4465 y Fu(n)638 4450 y Ft(<)29 b(b)784 4465 y Fu(n)832 4450 y Fx(.)47 b(Let)34 b Ft(b)1123 4465 y Fu(n)p Fm(+1)1290 4450 y Fx(:=)c Ft(b)1464 4465 y Fu(n)1511 4450 y Fx(.)47 b(Since)34 b Fn(Q)51 b Fx(is)33 b(dense)j(in)d Fn(R)5 b Fx(,)40 b(w)m(e)34 b(can)g(\014nd)h(a)e(rational)e Ft(a)3585 4465 y Fu(n)p Fm(+1)38 4566 y Fx(suc)m(h)j(that)936 4690 y(max)1118 4609 y Fj(\000)1164 4690 y Ft(x)1219 4705 y Fu(n)1266 4690 y Ft(;)17 b(a)1361 4705 y Fu(n)1408 4690 y Ft(;)g(b)1493 4705 y Fu(n)1562 4690 y Fr(\000)23 b Fx(2)1711 4649 y Fv(\000)p Fu(n)p Fv(\000)p Fm(1)1903 4609 y Fj(\001)1976 4690 y Ft(<)k(a)2130 4705 y Fu(n)p Fm(+1)2295 4690 y Ft(<)h(b)2440 4705 y Fu(n)p Fm(+1)2605 4690 y Fx(=)g Ft(b)2750 4705 y Fu(n)2797 4690 y Ft(:)137 4834 y Fw(Case)40 b Fx(2.)j Ft(a)552 4849 y Fu(n)627 4834 y Ft(<)27 b(x)785 4849 y Fu(n)833 4834 y Fx(.)43 b(Let)33 b Ft(a)1129 4849 y Fu(n)p Fm(+1)1294 4834 y Fx(:=)28 b Ft(a)1476 4849 y Fu(n)1523 4834 y Fx(,)k(and)h(\014nd)g(a)f (rational)e Ft(b)2452 4849 y Fu(n)p Fm(+1)2623 4834 y Fx(suc)m(h)k(that)942 4999 y Ft(a)993 5014 y Fu(n)1068 4999 y Fx(=)27 b Ft(a)1222 5014 y Fu(n)p Fm(+1)1387 4999 y Ft(<)h(b)1532 5014 y Fu(n)p Fm(+1)1697 4999 y Ft(<)g Fx(min)1963 4918 y Fj(\000)2009 4999 y Ft(x)2064 5014 y Fu(n)2111 4999 y Ft(;)17 b(b)2196 5014 y Fu(n)2243 4999 y Ft(;)g(a)2338 5014 y Fu(n)2407 4999 y Fx(+)22 b(2)2554 4958 y Fv(\000)p Fu(n)p Fv(\000)p Fm(1)2746 4918 y Fj(\001)2792 4999 y Ft(:)137 5165 y Fx(Clearly)30 b(\(1\))o(-\(3\))e(con)m(tin)m(ue)i(to)f(hold)f(for)g Ft(n)15 b Fx(+)g(1)28 b(\(as)h(far)f(as)h(de\014ned\).)44 b Ft(y)31 b Fx(:=)d(\()p Ft(a)3011 5180 y Fu(n)3058 5165 y Fx(\))3096 5180 y Fu(n)3171 5165 y Fx(is)h(a)f Fs(Ca)n(uchy)38 5282 y Fx(sequence,)42 b(since)d(for)e Ft(m)h Fr(\025)f Ft(n)h Fx(w)m(e)i(ha)m(v)m(e)f Fr(j)p Ft(a)1667 5297 y Fu(m)1759 5282 y Fr(\000)27 b Ft(a)1914 5297 y Fu(n)1961 5282 y Fr(j)37 b Fx(=)g Ft(a)2190 5297 y Fu(m)2282 5282 y Fr(\000)27 b Ft(a)2437 5297 y Fu(n)2521 5282 y Fr(\024)38 b Ft(b)2677 5297 y Fu(n)2750 5282 y Fr(\000)26 b Ft(a)2904 5297 y Fu(n)2989 5282 y Fr(\024)37 b Fx(2)3152 5245 y Fv(\000)p Fu(n)3254 5282 y Fx(.)60 b(Similarly)38 5398 y Ft(z)36 b Fx(:=)31 b(\()p Ft(b)332 5413 y Fu(n)379 5398 y Fx(\))417 5413 y Fu(n)499 5398 y Fx(is)j(a)h Fs(Ca)n(uchy)i Fx(sequence.)52 b Ft(y)35 b Fx(=)c Ft(z)39 b Fx(follo)m(ws)33 b(from)h(\(3\),)h(and)g(from)e(\(2\))h(together)h(with)38 5514 y(\(1\))d(w)m(e)i(obtain)d Ft(x)697 5529 y Fu(n)772 5514 y Ft(<)d(y)d Fr(_)e Ft(z)32 b(<)27 b(x)1273 5529 y Fu(n)1321 5514 y Fx(.)p 3682 5455 V 3682 5511 4 56 v 3720 5511 V 3682 5514 41 4 v 137 5690 a(After)g(some)e(computations)g (in)m(v)m(olving)g(real)g(n)m(um)m(b)s(ers)i(it)e(is)g(to)h(b)s(e)g (exp)s(ected)i(that)e(the)g(rational)38 5806 y(n)m(um)m(b)s(ers)f(o)s (ccurring)f(in)f(the)i Fs(Ca)n(uchy)i Fx(sequences)g(ma)m(y)d(b)s (ecome)h(rather)f(complex.)40 b(Hence)26 b(under)38 5922 y(computational)k(asp)s(ects)k(it)d(is)h(necessary)j(to)d(b)s(e)h(able) f(to)g Fw(cle)-5 b(an)34 b(up)39 b Fx(a)32 b(real,)g(as)h(follo)m(ws.) 38 6062 y Fp(Lemma)57 b(1.17.)51 b Fw(F)-7 b(or)50 b(every)h(r)-5 b(e)g(al)51 b Ft(x)58 b Fx(=)f(\(\()p Ft(a)1783 6077 y Fu(n)1830 6062 y Fx(\))1868 6077 y Fu(n)1915 6062 y Ft(;)17 b(M)10 b Fx(\))52 b Fw(we)e(c)-5 b(an)51 b(c)-5 b(onstruct)51 b(an)g(e)-5 b(quivalent)50 b(r)-5 b(e)g(al)38 6179 y Ft(y)35 b Fx(=)e(\(\()p Ft(b)347 6194 y Fu(n)394 6179 y Fx(\))432 6194 y Fu(n)479 6179 y Ft(;)17 b(N)10 b Fx(\))37 b Fw(wher)-5 b(e)37 b(the)h(r)-5 b(ationals)37 b Ft(b)1575 6194 y Fu(n)1659 6179 y Fw(ar)-5 b(e)38 b(of)f(the)g(form)g Ft(c)2389 6194 y Fu(n)2436 6179 y Ft(=)p Fx(2)2534 6143 y Fu(n)2618 6179 y Fw(with)g(inte)-5 b(gers)37 b Ft(c)3239 6194 y Fu(n)3286 6179 y Fw(,)h(and)f(with)38 6295 y(mo)-5 b(dulus)34 b Ft(N)10 b Fx(\()p Ft(k)s Fx(\))29 b(=)e Ft(k)e Fx(+)d(2)p Fw(.)p eop %%Page: 6 6 6 5 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(c)553 465 y Fu(n)627 450 y Fx(:=)28 b Fr(b)p Ft(a)853 466 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))1052 450 y Fr(\001)22 b Fx(2)1151 414 y Fu(n)1198 450 y Fr(c)33 b Fx(and)f Ft(b)1505 465 y Fu(n)1580 450 y Fx(:=)c Ft(c)1753 465 y Fu(n)1822 450 y Fr(\001)22 b Fx(2)1921 414 y Fv(\000)p Fu(n)2023 450 y Fx(,)32 b(hence)1102 596 y Ft(c)1144 611 y Fu(n)p 1099 640 96 4 v 1099 732 a Fx(2)1148 703 y Fu(n)1232 663 y Fr(\024)c Ft(a)1388 679 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))1593 663 y Ft(<)1710 596 y(c)1752 611 y Fu(n)p 1706 640 V 1706 732 a Fx(2)1755 703 y Fu(n)1834 663 y Fx(+)1966 596 y(1)p 1942 640 V 1942 732 a(2)1991 703 y Fu(n)2145 663 y Fx(with)33 b Ft(c)2410 678 y Fu(n)2484 663 y Fr(2)28 b Fn(Z)p Fx(.)38 850 y(Then)34 b(for)e Ft(m)c Fr(\024)g Ft(n)435 995 y Fr(j)p Ft(b)504 1010 y Fu(m)593 995 y Fr(\000)22 b Ft(b)733 1010 y Fu(n)781 995 y Fr(j)27 b Fx(=)h Fr(j)p Ft(c)1010 1010 y Fu(m)1098 995 y Fr(\001)22 b Fx(2)1197 954 y Fv(\000)p Fu(m)1340 995 y Fr(\000)h Ft(c)1482 1010 y Fu(n)1551 995 y Fr(\001)f Fx(2)1650 954 y Fv(\000)p Fu(n)1751 995 y Fr(j)836 1152 y(\024)28 b(j)p Ft(c)1011 1167 y Fu(m)1100 1152 y Fr(\001)21 b Fx(2)1198 1111 y Fv(\000)p Fu(m)1342 1152 y Fr(\000)h Ft(a)1492 1168 y Fu(M)7 b Fm(\()p Fu(m)p Fm(\))1689 1152 y Fr(j)22 b Fx(+)g Fr(j)p Ft(a)1916 1168 y Fu(M)7 b Fm(\()p Fu(m)p Fm(\))2134 1152 y Fr(\000)23 b Ft(a)2285 1168 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))2462 1152 y Fr(j)21 b Fx(+)h Fr(j)p Ft(a)2688 1168 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))2887 1152 y Fr(\000)23 b Ft(c)3029 1167 y Fu(n)3098 1152 y Fr(\001)f Fx(2)3197 1111 y Fv(\000)p Fu(n)3298 1152 y Fr(j)836 1310 y(\024)28 b Fx(2)990 1269 y Fv(\000)p Fu(m)1134 1310 y Fx(+)22 b(2)1281 1269 y Fv(\000)p Fu(m)1424 1310 y Fx(+)g(2)1571 1269 y Fv(\000)p Fu(n)836 1470 y Ft(<)28 b Fx(2)989 1429 y Fv(\000)p Fu(m)p Fm(+2)1200 1470 y Ft(;)38 1641 y Fx(hence)k Fr(j)p Ft(b)376 1656 y Fu(m)460 1641 y Fr(\000)18 b Ft(b)596 1656 y Fu(n)643 1641 y Fr(j)28 b(\024)g Fx(2)853 1605 y Fv(\000)p Fu(k)981 1641 y Fx(for)h Ft(n)f Fr(\025)g Ft(m)g Fr(\025)h Ft(k)20 b Fx(+)e(2)27 b(=:)h Ft(N)10 b Fx(\()p Ft(k)s Fx(\),)31 b(so)g(\()p Ft(b)2382 1656 y Fu(n)2429 1641 y Fx(\))2467 1656 y Fu(n)2544 1641 y Fx(is)f(a)g Fs(Ca)n(uchy)j Fx(sequence)g(with) 38 1758 y(mo)s(dulus)e Ft(N)10 b Fx(.)137 1874 y(T)-8 b(o)33 b(pro)m(v)m(e)h(that)e Ft(x)h Fx(is)f(equiv)-5 b(alen)m(t)33 b(to)f Ft(y)f Fx(:=)c(\(\()p Ft(b)1850 1889 y Fu(n)1898 1874 y Fx(\))1936 1889 y Fu(n)1983 1874 y Ft(;)17 b(N)10 b Fx(\),)32 b(observ)m(e)816 2029 y Fr(j)p Ft(a)895 2044 y Fu(n)964 2029 y Fr(\000)23 b Ft(b)1105 2044 y Fu(n)1153 2029 y Fr(j)k(\024)h(j)p Ft(a)1392 2044 y Fu(n)1461 2029 y Fr(\000)23 b Ft(a)1612 2045 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))1789 2029 y Fr(j)21 b Fx(+)h Fr(j)p Ft(a)2015 2045 y Fu(M)7 b Fm(\()p Fu(n)p Fm(\))2214 2029 y Fr(\000)23 b Ft(c)2356 2044 y Fu(n)2425 2029 y Fr(\001)f Fx(2)2524 1988 y Fv(\000)p Fu(n)2625 2029 y Fr(j)1208 2193 y(\024)28 b Fx(2)1362 2152 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1572 2193 y Fx(+)22 b(2)1719 2152 y Fv(\000)p Fu(n)1918 2193 y Fx(for)32 b Ft(n;)17 b(M)10 b Fx(\()p Ft(n)p Fx(\))28 b Fr(\025)h Ft(M)10 b Fx(\()p Ft(k)25 b Fx(+)d(1\))1208 2357 y Fr(\024)28 b Fx(2)1362 2316 y Fv(\000)p Fu(k)1557 2357 y Fx(if)j(in)h(addition)f Ft(n)d Fr(\025)g Ft(k)d Fx(+)d(1.)38 2528 y(Hence)34 b Fr(j)p Ft(a)407 2543 y Fu(n)476 2528 y Fr(\000)22 b Ft(b)616 2543 y Fu(n)664 2528 y Fr(j)27 b(\024)h Fx(2)873 2491 y Fv(\000)p Fu(k)1003 2528 y Fx(for)k Ft(n)c Fr(\025)g Fx(max\()p Ft(k)d Fx(+)d(1)p Ft(;)17 b(M)10 b Fx(\()p Ft(k)25 b Fx(+)d(1\)\),)33 b(and)f(therefore)h Ft(x)c Fx(=)e Ft(y)t Fx(.)p 3682 2469 41 4 v 3682 2524 4 56 v 3720 2524 V 3682 2527 41 4 v 819 2727 a(2.)49 b Fs(Sequences)36 b(and)i(Series)e(of)j(Real)e(Numbers)38 2901 y Fp(De\014nition)52 b(2.1.)d Fx(A)e(sequence)j(\()p Ft(x)1432 2916 y Fu(n)1479 2901 y Fx(\))1517 2916 y Fu(n)p Fv(2)p Fo(N)1706 2901 y Fx(of)c(reals)h(is)f(a)h Fs(Ca)n(uchy)i Fw(se)-5 b(quenc)g(e)53 b Fx(with)47 b(mo)s(dulus)38 3017 y Ft(M)22 b Fx(:)34 b Fn(N)48 b Fr(!)32 b Fn(N)50 b Fx(whenev)m(er)38 b Fr(j)p Ft(x)1073 3032 y Fu(m)1164 3017 y Fr(\000)25 b Ft(x)1321 3032 y Fu(n)1368 3017 y Fr(j)33 b(\024)g Fx(2)1588 2981 y Fv(\000)p Fu(k)1721 3017 y Fx(for)i Ft(m;)17 b(n)32 b Fr(\025)i Ft(M)10 b Fx(\()p Ft(k)s Fx(\),)37 b(and)e Fw(c)-5 b(onver)g(ges)43 b Fx(with)35 b(mo)s(dulus)38 3134 y Ft(M)22 b Fx(:)33 b Fn(N)43 b Fr(!)27 b Fn(N)47 b Fx(to)33 b(a)f(real)g Ft(y)t Fx(,)f(its)h Fw(limit)p Fx(,)h(whenev)m(er)i Fr(j)p Ft(x)1950 3149 y Fu(n)2019 3134 y Fr(\000)23 b Ft(y)t Fr(j)j(\024)i Fx(2)2379 3097 y Fv(\000)p Fu(k)2509 3134 y Fx(for)k Ft(n)c Fr(\025)g Ft(M)10 b Fx(\()p Ft(k)s Fx(\).)137 3271 y(Clearly)32 b(the)h(limit)c(of)j(a)h(con)m(v)m(ergen)m (t)h(sequence)i(of)c(reals)g(is)g(uniqely)g(determined.)38 3407 y Fp(Lemma)i(2.2.)40 b Fw(Every)33 b(mo)-5 b(dulate)g(d)42 b Fs(Ca)n(uchy)33 b Fw(se)-5 b(quenc)g(e)32 b(of)g(r)-5 b(ationals)32 b(c)-5 b(onver)g(ges)32 b(with)g(the)h(same)38 3524 y(mo)-5 b(dulus)34 b(to)h(the)g(r)-5 b(e)g(al)35 b(numb)-5 b(er)34 b(it)h(r)-5 b(epr)g(esents.)38 3690 y(Pr)g(o)g(of.)41 b Fx(Let)f Ft(x)i Fx(:=)f(\(\()p Ft(a)886 3705 y Fu(n)933 3690 y Fx(\))971 3705 y Fu(n)1018 3690 y Ft(;)17 b(M)10 b Fx(\))40 b(b)s(e)h(a)f(real.)66 b(W)-8 b(e)41 b(m)m(ust)f(sho)m(w)i Fr(j)p Ft(a)2479 3705 y Fu(n)2553 3690 y Fr(\000)28 b Ft(x)p Fr(j)41 b(\024)g Fx(2)2949 3654 y Fv(\000)p Fu(k)3087 3690 y Fx(for)f Ft(n)h Fr(\025)g Ft(M)10 b Fx(\()p Ft(k)s Fx(\).)38 3807 y(Fix)36 b Ft(n)g Fr(\025)f Ft(M)10 b Fx(\()p Ft(k)s Fx(\).)58 b(It)37 b(su\016ces)i(to)e(sho)m(w)h Fr(j)p Ft(a)1637 3822 y Fu(m)1729 3807 y Fr(\000)25 b Ft(a)1882 3822 y Fu(n)1930 3807 y Fr(j)35 b(\024)g Fx(2)2154 3770 y Fv(\000)p Fu(k)2289 3807 y Fx(for)h Ft(n)g Fr(\025)g Ft(M)10 b Fx(\()p Ft(k)s Fx(\).)57 b(But)37 b(this)g(holds)g(b)m(y)38 3923 y(assumption.)p 3682 3864 V 3682 3919 4 56 v 3720 3919 V 3682 3922 41 4 v 137 4090 a(By)g(the)g(triangle)d(inequalit)m(y) -8 b(,)36 b(ev)m(ery)i(con)m(v)m(ergen)m(t)g(sequence)h(of)c(reals)h (with)g(mo)s(dulus)f Ft(M)46 b Fx(is)36 b(a)38 4206 y Fs(Ca)n(uchy)30 b Fx(sequence)h(with)c(mo)s(dulus)g Ft(k)j Fr(7!)e Ft(M)10 b Fx(\()p Ft(k)16 b Fx(+)c(1\).)42 b(W)-8 b(e)28 b(no)m(w)h(pro)m(v)m(e)g(the)f(rev)m(erse)i(implication.)38 4343 y Fp(Theorem)g(2.3)h Fx(\(Sequen)m(tial)c(Completeness\))p Fp(.)38 b Fw(F)-7 b(or)29 b(every)39 b Fs(Ca)n(uchy)30 b Fw(se)-5 b(quenc)g(e)28 b(of)i(r)-5 b(e)g(als)29 b(we)g(c)-5 b(an)38 4459 y(\014nd)34 b(a)h(r)-5 b(e)g(al)34 b(to)h(which)f(it)h(c) -5 b(onver)g(ges.)38 4626 y(Pr)g(o)g(of.)41 b Fx(Let)k(\()p Ft(x)616 4641 y Fu(n)664 4626 y Fx(\))702 4641 y Fu(n)p Fv(2)p Fo(N)889 4626 y Fx(b)s(e)g(a)g Fs(Ca)n(uchy)i Fx(sequence)h(of)d(reals)g(with)g(mo)s(dulus)f Ft(M)10 b Fx(;)52 b(assume)46 b Ft(x)3565 4641 y Fu(n)3657 4626 y Fx(is)38 4742 y(\(\()p Ft(a)165 4757 y Fu(nk)250 4742 y Fx(\))288 4757 y Fu(k)331 4742 y Ft(;)17 b(N)453 4757 y Fu(n)500 4742 y Fx(\).)81 b(Note)46 b(\014rst)g(that,)i(for)d(eac)m (h)h Ft(n)k Fr(2)g Fn(N)60 b Fx(and)45 b(ev)m(ery)i Ft(p)p Fx(,)i(b)m(y)d(Lemma)e(2.2)h(w)m(e)h(ha)m(v)m(e)38 4858 y Fr(j)p Ft(x)121 4873 y Fu(n)190 4858 y Fr(\000)23 b Ft(a)341 4873 y Fu(nl)410 4858 y Fr(j)k(\024)h Fx(2)619 4822 y Fv(\000)p Fu(p)746 4858 y Fx(for)k(all)f Ft(l)e Fr(\025)g Ft(N)1273 4873 y Fu(n)1320 4858 y Fx(\()p Ft(p)p Fx(\).)43 b(Next,)34 b(set)1623 5016 y Ft(b)1664 5031 y Fu(n)1739 5016 y Fx(=)28 b Ft(a)1894 5031 y Fu(nN)1993 5039 y Fk(n)2035 5031 y Fm(\()p Fu(n)p Fm(\))38 5174 y Fx(for)k(ev)m(ery)i Ft(n)28 b Fr(2)h Fn(N)8 b Fx(,)39 b(so)33 b(that)f Fr(j)p Ft(x)1170 5189 y Fu(n)1240 5174 y Fr(\000)22 b Ft(b)1380 5189 y Fu(n)1428 5174 y Fr(j)27 b(\024)h Fx(2)1637 5137 y Fv(\000)p Fu(n)1772 5174 y Fx(for)k(all)e Ft(n)e Fr(2)g Fn(N)48 b Fx(b)m(y)34 b(the)f(particular)d (case)k Ft(l)c Fx(=)e Ft(N)3542 5189 y Fu(n)3589 5174 y Fx(\()p Ft(n)p Fx(\))38 5290 y(of)k(the)h(foregoing)e(consideration.) 43 b(Then)318 5447 y Fr(j)p Ft(b)387 5462 y Fu(m)476 5447 y Fr(\000)23 b Ft(b)617 5462 y Fu(n)664 5447 y Fr(j)28 b(\024)g(j)p Ft(b)894 5462 y Fu(m)983 5447 y Fr(\000)22 b Ft(x)1137 5462 y Fu(m)1204 5447 y Fr(j)g Fx(+)g Fr(j)p Ft(x)1435 5462 y Fu(m)1524 5447 y Fr(\000)h Ft(x)1679 5462 y Fu(n)1726 5447 y Fr(j)f Fx(+)g Fr(j)p Ft(x)1957 5462 y Fu(n)2026 5447 y Fr(\000)h Ft(b)2167 5462 y Fu(n)2214 5447 y Fr(j)k(\024)i Fx(2)2424 5406 y Fv(\000)p Fu(m)2567 5447 y Fx(+)22 b(2)2714 5406 y Fv(\000)p Fu(q)r Fv(\000)p Fm(1)2919 5447 y Fx(+)g(2)3066 5406 y Fv(\000)p Fu(n)3195 5447 y Fr(\024)29 b Fx(2)3350 5406 y Fv(\000)p Fu(q)38 5605 y Fx(for)42 b(all)f Ft(m;)17 b(n)46 b Fr(\025)g Fx(max\()p Ft(M)10 b Fx(\()p Ft(q)34 b Fx(+)29 b(1\))p Ft(;)17 b(q)32 b Fx(+)e(2\),)45 b(whic)m(h)f(is)e(to)h(sa)m(y)h(that)f Ft(y)49 b Fx(:=)d(\()p Ft(b)2968 5620 y Fu(n)3015 5605 y Fx(\))3053 5620 y Fu(n)3143 5605 y Fx(is)d(a)f Fs(Ca)n(uchy)38 5721 y Fx(sequence)35 b(with)d(mo)s(dulus)g Ft(L)p Fx(\()p Ft(q)t Fx(\))27 b(:=)h(max\()p Ft(M)10 b Fx(\()p Ft(q)26 b Fx(+)c(1\))p Ft(;)17 b(q)26 b Fx(+)c(2\).)43 b(Moreo)m(v)m(er,)34 b(again)d(b)m(y)j(Lemma)d(2.2)791 5879 y Fr(j)p Ft(x)874 5894 y Fu(n)943 5879 y Fr(\000)23 b Ft(y)t Fr(j)j(\024)j(j)p Ft(x)1338 5894 y Fu(n)1407 5879 y Fr(\000)22 b Ft(b)1547 5894 y Fu(n)1595 5879 y Fr(j)g Fx(+)g Fr(j)p Ft(b)1812 5894 y Fu(n)1881 5879 y Fr(\000)h Ft(y)t Fr(j)j(\024)i Fx(2)2241 5838 y Fv(\000)p Fu(n)2365 5879 y Fx(+)22 b(2)2512 5838 y Fv(\000)p Fu(q)r Fv(\000)p Fm(1)2723 5879 y Fr(\024)28 b Fx(2)2877 5838 y Fv(\000)p Fu(q)38 6037 y Fx(for)e(all)e Ft(n)k Fr(\025)g Ft(L)p Fx(\()p Ft(q)13 b Fx(+)d(1\);)29 b(in)c(other)i(w)m(ords:)41 b(\()p Ft(x)1653 6052 y Fu(n)1701 6037 y Fx(\))26 b(con)m(v)m(erges)j(to)d Ft(y)j Fx(with)e(mo)s(dulus)e Ft(q)31 b Fr(7!)d Ft(L)p Fx(\()p Ft(q)13 b Fx(+)d(1\).)p 3682 5978 V 3682 6034 4 56 v 3720 6034 V 3682 6037 41 4 v 137 6204 a(It)37 b(will)c(b)s(e)k(useful)f(to)f(c)m(haracterize)i (con)m(v)m(ergence)i(of)d(a)f(sequence)k(of)d(reals)f(in)h(terms)g(of)f (their)38 6320 y(appro)m(ximations.)p eop %%Page: 7 7 7 6 bop 38 450 a Fp(Lemma)37 b(2.4.)42 b Fw(F)-7 b(or)34 b(r)-5 b(e)g(als)34 b Ft(x)1113 465 y Fu(i)1142 450 y Ft(;)17 b(y)37 b Fw(r)-5 b(epr)g(esente)g(d)35 b(by)g Fx(\()p Ft(a)1996 465 y Fu(i;n)2086 450 y Fx(\))2124 465 y Fu(n)2171 450 y Ft(;)17 b Fx(\()p Ft(b)2294 465 y Fu(n)2341 450 y Fx(\))2379 465 y Fu(n)2461 450 y Fw(the)35 b(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)38 599 y Fx(\(a\))41 b(\()p Ft(x)297 614 y Fu(i)326 599 y Fx(\))364 614 y Fu(i)426 599 y Fw(c)-5 b(onver)g(ges)34 b(to)h Ft(y)t Fw(,)f(i.e.,)g Fr(8)p Ft(k)s Fr(9)p Ft(q)t Fr(8)p Ft(i)p Fr(\025)r Ft(q)e Fx(\()p Fr(j)p Ft(x)1867 614 y Fu(i)1917 599 y Fr(\000)23 b Ft(y)t Fr(j)j(\024)i Fx(2)2277 563 y Fv(\000)p Fu(k)2374 599 y Fx(\))p Fw(;)32 716 y Fx(\(b\))42 b Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(i;)17 b(n)p Fr(\025)r Ft(p)27 b Fx(\()p Fr(j)p Ft(a)879 731 y Fu(in)972 716 y Fr(\000)c Ft(b)1113 731 y Fu(n)1160 716 y Fr(j)28 b(\024)g Fx(2)1370 679 y Fv(\000)p Fu(k)1467 716 y Fx(\))p Fw(.)38 907 y(Pr)-5 b(o)g(of.)41 b Fx(\(a\))31 b Fr(\))h Fx(\(b\).)43 b(Giv)m(en)32 b Ft(k)s Fx(,)g(pic)m(k)g Ft(q)k Fx(b)m(y)e(\(a\))e(suc)m(h)h(that)f Fr(j)p Ft(x)2309 922 y Fu(i)2358 907 y Fr(\000)21 b Ft(y)t Fr(j)26 b(\024)j Fx(2)2717 871 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2936 907 y Fx(for)i Ft(i)d(>)g(q)t Fx(.)43 b(Then)33 b(b)m(y)38 1023 y(de\014nition)e Fr(j)p Ft(a)550 1038 y Fu(in)643 1023 y Fr(\000)23 b Ft(b)784 1038 y Fu(n)831 1023 y Fr(j)28 b(\024)g Fx(2)1041 987 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1251 1023 y Fx(+)22 b(2)1398 987 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1618 1023 y Fx(for)32 b Ft(n)c Fr(\025)g Ft(r)35 b Fx(with)d Ft(r)j Fx(dep)s(ending)e(on)g Ft(k)i Fx(only)-8 b(.)137 1139 y(\(b\))29 b Fr(\))g Fx(\(a\))o(.)43 b(Giv)m(en)28 b Ft(k)s Fx(,)i(w)m(e)g(ha)m(v)m(e)g(to)f(\014nd)g Ft(q)k Fx(suc)m(h)d(that)f Fr(j)p Ft(x)2259 1154 y Fu(i)2302 1139 y Fr(\000)15 b Ft(y)t Fr(j)26 b(\024)i Fx(2)2654 1103 y Fv(\000)p Fu(k)2780 1139 y Fx(for)h Ft(i)f(>)f(q)t Fx(.)42 b(It)29 b(su\016ces)i(to)38 1256 y(ha)m(v)m(e)k Fr(j)p Ft(a)343 1271 y Fu(in)437 1256 y Fr(\000)23 b Ft(b)578 1271 y Fu(n)626 1256 y Fr(j)29 b(\024)i Fx(2)840 1220 y Fv(\000)p Fu(k)960 1256 y Fx(+)23 b(2)1108 1220 y Fv(\000)p Fu(l)1222 1256 y Fx(for)34 b Ft(n)c Fr(\025)g Ft(r)37 b Fx(with)c Ft(r)k Fx(dep)s(ending)d(on)f Ft(l)r Fx(.)48 b(But)34 b(b)m(y)i(\(b\))e(w)m(e)h(ev)m(en)g(ha)m(v)m(e)38 1372 y Fr(j)p Ft(a)117 1387 y Fu(in)210 1372 y Fr(\000)22 b Ft(b)350 1387 y Fu(n)398 1372 y Fr(j)27 b(\024)h Fx(2)607 1336 y Fv(\000)p Fu(k)737 1372 y Fx(for)k Ft(n)c Fr(\025)g Ft(p)p Fx(.)p 3682 1313 41 4 v 3682 1369 4 56 v 3720 1369 V 3682 1372 41 4 v 1252 1629 a(3.)48 b Fs(Continuous)39 b(Functions)137 1803 y Fx(An)c(inhabited,)e(closed)h(\014nite)g(in)m (terv)-5 b(al)32 b(is)i(called)e(a)i Fw(c)-5 b(omp)g(act)35 b(interval)10 b Fx(.)48 b(By)34 b(default)f(w)m(e)i(shall)38 1919 y(assume)e(that)f(a)h(compact)f(in)m(terv)-5 b(al)31 b Ft(I)36 b Fx(=)27 b([)p Ft(a;)17 b(b)p Fx(])33 b(has)g(rational)d (end-p)s(oin)m(ts.)38 2066 y Fp(De\014nition)39 b(3.1.)44 b Fx(Let)35 b Ft(S)41 b Fx(b)s(e)36 b(a)f(set)h(of)f(reals.)52 b(A)36 b(real)e Ft(y)39 b Fx(is)c(an)g Fw(upp)-5 b(er)38 b(b)-5 b(ound)45 b Fx(of)35 b Ft(S)41 b Fx(if)35 b Ft(x)e Fr(\024)g Ft(y)38 b Fx(for)38 2183 y(all)d Ft(x)g Fr(2)h Ft(S)6 b Fx(.)57 b(A)37 b(real)f Ft(y)k Fx(is)d(a)g Fw(supr)-5 b(emum)44 b Fx(of)36 b Ft(S)43 b Fx(if)36 b Ft(y)k Fx(is)d(an)g(upp)s (er)g(b)s(ound)h(of)e Ft(S)6 b Fx(,)38 b(and)f(in)g(addition)38 2299 y(for)e(ev)m(ery)i(rational)c Ft(a)f(<)g(y)39 b Fx(there)d(is)f(real)f Ft(x)f Fr(2)g Ft(S)41 b Fx(suc)m(h)36 b(that)g Ft(a)c Fr(\024)h Ft(x)p Fx(.)52 b(A)35 b(set)i Ft(S)k Fx(of)34 b(reals)h(is)g Fw(or)-5 b(der)38 2415 y(lo)g(c)g(ate)g(d)33 b(ab)-5 b(ove)38 b Fx(if)30 b(for)h(ev)m(ery)i Ft(a)28 b(<)g(b)p Fx(,)k(either)f Ft(x)d Fr(\024)h Ft(b)i Fx(for)g(all)f Ft(x)e Fr(2)g Ft(S)37 b Fx(or)31 b(else)h Ft(a)c Fr(\024)g Ft(x)k Fx(for)e(some)i Ft(x)c Fr(2)g Ft(S)6 b Fx(.)137 2562 y(Ev)m(ery)48 b(set)e Ft(S)52 b Fx(can)46 b(ha)m(v)m(e)h(at)e(most)g(one)h(suprem)m(um.)83 b(T)-8 b(o)46 b(see)h(this,)i(assume)d(that)g Ft(y)t(;)17 b(z)49 b Fx(are)38 2678 y(suprema)37 b(of)g Ft(S)6 b Fx(.)57 b(It)37 b(is)g(enough)g(to)g(sho)m(w)h Ft(y)h Fr(\024)d Ft(z)t Fx(,)j(and)e(for)g(this)g(it)f(su\016ces)j(to)e(sho)m (w)h Ft(z)i Fr(6)p Ft(<)c(y)t Fx(.)56 b(So)38 2795 y(assume)42 b Ft(z)47 b(<)42 b(y)t Fx(.)68 b(Then)43 b Ft(z)k(<)42 b(a)h(<)f(y)i Fx(for)d(some)g(rational)d Ft(a)p Fx(,)44 b(hence)f Ft(a)f Fr(\024)h Ft(x)f Fx(for)e(some)h Ft(x)i Fr(2)g Ft(S)6 b Fx(,)38 2911 y(con)m(tradicting)39 b(the)i(assumption)f (that)h Ft(z)k Fx(is)40 b(an)g(upp)s(er)i(b)s(ound)e(of)g Ft(S)6 b Fx(.)68 b(If)40 b(the)h(suprem)m(um)g(of)f Ft(S)38 3027 y Fx(exists,)33 b(it)f(is)g(denoted)h(b)m(y)h(sup)17 b Ft(S)6 b Fx(.)38 3174 y Fp(Theorem)45 b(3.2)h Fx(\(Least-Upp)s (er-Bound)39 b(Principle\))p Fp(.)44 b Fw(Assume)e(that)f Ft(S)47 b Fw(is)41 b(an)g(inhabite)-5 b(d)41 b(set)g(of)38 3290 y(r)-5 b(e)g(als)34 b(that)h(is)g(b)-5 b(ounde)g(d)34 b(ab)-5 b(ove.)44 b(Then)34 b Ft(S)41 b Fw(has)34 b(a)h(supr)-5 b(emum)34 b(i\013)h(it)g(is)g(or)-5 b(der)34 b(lo)-5 b(c)g(ate)g(d)34 b(ab)-5 b(ove.)38 3482 y(Pr)g(o)g(of.)41 b Fx(If)32 b(sup)18 b Ft(S)38 b Fx(exists)33 b(and)f Ft(a)c(<)g(b)p Fx(,)33 b(then)f(either)h(sup)17 b Ft(S)34 b(<)27 b(b)33 b Fx(or)f(else)g Ft(a)c(<)g Fx(sup)17 b Ft(S)6 b Fx(.)43 b(In)33 b(the)g(former)38 3598 y(case)i Ft(x)30 b Fr(\024)g Ft(b)k Fx(for)f(all)f Ft(x)e Fr(2)h Ft(S)6 b Fx(,)34 b(and)g(in)f(the)h(latter)f(case)h(clearly)f Ft(a)d Fr(\024)g Ft(x)35 b Fx(for)e(some)h Ft(x)c Fr(2)g Ft(S)6 b Fx(.)47 b(Hence)35 b Ft(S)38 3714 y Fx(is)d(order)h(lo)s (cated)e(ab)s(o)m(v)m(e.)137 3830 y(F)-8 b(or)32 b(the)h(con)m(v)m (erse)i(it)d(is)g(useful)g(to)h(consider)509 4008 y(\005)582 4023 y Fu(S)633 4008 y Fx(\()p Ft(a;)17 b(b)p Fx(\):)239 b(b)s(oth)32 b Ft(y)f Fr(\024)d Ft(b)33 b Fx(for)f(all)f Ft(y)g Fr(2)d Ft(S)38 b Fx(and)33 b Ft(a)27 b(<)h(x)33 b Fx(for)f(some)g Ft(x)d Fr(2)f Ft(S)38 4186 y Fx(as)34 b(a)g(prop)s(ert)m(y)h(of)e(an)m(y)i(pair)e Ft(a;)17 b(b)35 b Fx(of)e(rational)f(n)m(um)m(b)s(ers)j(with)f Ft(a)c(<)h(b)p Fx(.)48 b(By)35 b(assumption)e(w)m(e)i(ha)m(v)m(e)38 4302 y Ft(a;)17 b(b)29 b Fr(2)g Fn(Q)50 b Fx(with)33 b Ft(a)c(<)f(b)34 b Fx(suc)m(h)g(that)f(\005)1395 4317 y Fu(S)1446 4302 y Fx(\()p Ft(a;)17 b(b)p Fx(\).)45 b(W)-8 b(e)34 b(construct)g(t)m(w)m(o)f(sequences)k(\()p Ft(c)3036 4317 y Fu(n)3083 4302 y Fx(\))3121 4317 y Fu(n)3201 4302 y Fx(and)c(\()p Ft(d)3480 4317 y Fu(n)3526 4302 y Fx(\))3564 4317 y Fu(n)3644 4302 y Fx(of)38 4418 y(rationals)d(suc)m(h)k(that)f (for)f(all)e Ft(n)844 4596 y(a)e Fx(=)f Ft(c)1068 4611 y Fm(0)1136 4596 y Fr(\024)h Ft(c)1283 4611 y Fm(1)1350 4596 y Fr(\024)g(\001)17 b(\001)g(\001)26 b(\024)i Ft(c)1746 4611 y Fu(n)1821 4596 y Ft(<)f(d)1975 4611 y Fu(n)2050 4596 y Fr(\024)h(\001)17 b(\001)g(\001)26 b(\024)i Ft(d)2455 4611 y Fm(1)2522 4596 y Fr(\024)g Ft(d)2678 4611 y Fm(0)2744 4596 y Fx(=)g Ft(b;)-2878 b Fx(\(4\))844 4747 y(\005)917 4762 y Fu(S)968 4747 y Fx(\()p Ft(c)1048 4762 y Fu(n)1095 4747 y Ft(;)17 b(d)1190 4762 y Fu(n)1236 4747 y Fx(\))p Ft(;)-1263 b Fx(\(5\))844 4955 y Ft(d)895 4970 y Fu(n)964 4955 y Fr(\000)22 b Ft(c)1105 4970 y Fu(n)1180 4955 y Fr(\024)1285 4844 y Fj(\020)1355 4888 y Fx(2)p 1355 4932 49 4 v 1355 5023 a(3)1413 4844 y Fj(\021)1473 4867 y Fu(n)1520 4955 y Fx(\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))p Ft(:)-1799 b Fx(\(6\))38 5168 y(Let)41 b Ft(c)263 5183 y Fm(0)303 5168 y Ft(;)17 b(:)g(:)g(:)e(;)i(c)563 5183 y Fu(n)651 5168 y Fx(and)42 b Ft(d)901 5183 y Fm(0)940 5168 y Ft(;)17 b(:)g(:)g(:)f(;)h(d)1210 5183 y Fu(n)1297 5168 y Fx(b)s(e)42 b(already)f(constructed)h(suc)m(h)h(that)e (\(4\)-\(6\))f(hold.)69 b(Let)42 b Ft(c)g Fx(=)38 5284 y Ft(c)80 5299 y Fu(n)151 5284 y Fx(+)260 5245 y Fm(1)p 260 5261 36 4 v 260 5318 a(3)305 5284 y Fx(\()p Ft(d)394 5299 y Fu(n)465 5284 y Fr(\000)24 b Ft(c)608 5299 y Fu(n)655 5284 y Fx(\))35 b(and)g Ft(d)d Fx(=)f Ft(c)1152 5299 y Fu(n)1223 5284 y Fx(+)1333 5245 y Fm(2)p 1333 5261 V 1333 5318 a(3)1378 5284 y Fx(\()p Ft(d)1467 5299 y Fu(n)1537 5284 y Fr(\000)24 b Ft(c)1680 5299 y Fu(n)1727 5284 y Fx(\).)51 b(Since)35 b Ft(S)41 b Fx(is)34 b(order)i(lo)s(cated)e (ab)s(o)m(v)m(e,)i(either)f Ft(s)d Fr(\024)g Ft(d)38 5400 y Fx(for)e(all)e Ft(s)g Fr(2)g Ft(S)36 b Fx(or)30 b(else)h Ft(c)d(<)f(r)34 b Fx(for)c(some)g Ft(r)g Fr(2)f Ft(S)6 b Fx(.)42 b(In)31 b(the)g(\014rst)g(case)g(let)f Ft(c)2696 5415 y Fu(n)p Fm(+1)2861 5400 y Fx(:=)e Ft(c)3034 5415 y Fu(n)3111 5400 y Fx(and)j Ft(d)3350 5415 y Fu(n)p Fm(+1)3514 5400 y Fx(:=)d Ft(d)p Fx(,)38 5516 y(and)36 b(in)f(the)i(second)g(case)g(let)f Ft(c)1235 5531 y Fu(n)p Fm(+1)1406 5516 y Fx(:=)d Ft(c)j Fx(and)g Ft(d)1864 5531 y Fu(n)p Fm(+1)2035 5516 y Fx(:=)e Ft(d)2223 5531 y Fu(n)2269 5516 y Fx(.)54 b(Then)38 b(clearly)d(\005)2997 5531 y Fu(S)3048 5516 y Fx(\()p Ft(c)3128 5531 y Fu(n)p Fm(+1)3265 5516 y Ft(;)17 b(d)3360 5531 y Fu(n)p Fm(+1)3496 5516 y Fx(\),)37 b(\(4\))38 5633 y(and)32 b(\(6\))g(con)m(tin)m(ue)h(to)f (hold)f(for)h Ft(n)21 b Fx(+)g(1,)33 b(and)f(the)g(real)g(n)m(um)m(b)s (er)g Ft(x)c Fx(=)g Ft(y)35 b Fx(giv)m(en)e(b)m(y)g(the)f(mo)s(dulated) 38 5749 y Fs(Ca)n(uchy)j Fx(sequences)g(of)e(rationals)d(\()p Ft(c)1487 5764 y Fu(n)1534 5749 y Fx(\))1572 5764 y Fu(n)1651 5749 y Fx(and)j(\()p Ft(d)1930 5764 y Fu(n)1977 5749 y Fx(\))2015 5764 y Fu(n)2094 5749 y Fx(is)f(the)h(least)f(upp)s(er)i (b)s(ound)e(of)g Ft(S)6 b Fx(.)p 3682 5690 41 4 v 3682 5746 4 56 v 3720 5746 V 3682 5749 41 4 v 38 5940 a Fp(De\014nition)29 b(3.3.)37 b Fx(A)26 b Fw(c)-5 b(ontinuous)29 b(function)k Ft(f)22 b Fx(:)33 b Ft(I)j Fr(!)27 b Fn(R)37 b Fx(on)26 b(a)g(compact)g(in)m(terv)-5 b(al)25 b Ft(I)34 b Fx(with)26 b(rational)38 6057 y(end)33 b(p)s(oin)m(ts)f(is)g(giv)m(en)h(b)m(y)38 6204 y(\(a\))41 b(an)g(appro)m(ximating)f(map)h Ft(h)1290 6219 y Fu(f)1346 6204 y Fx(:)c(\()p Ft(I)f Fr(\\)28 b Fn(Q)12 b Fx(\))34 b Fr(\002)29 b Fn(N)58 b Fr(!)42 b Fn(Q)59 b Fx(and)42 b(a)f(map)g Ft(\013)2824 6219 y Fu(f)2881 6204 y Fx(:)36 b Fn(N)58 b Fr(!)42 b Fn(N)57 b Fx(suc)m(h)43 b(that)204 6320 y(\()p Ft(h)298 6335 y Fu(f)343 6320 y Fx(\()p Ft(a;)17 b(n)p Fx(\)\))610 6335 y Fu(n)690 6320 y Fx(is)32 b(a)g Fs(Ca)n(uchy)j Fx(sequence)g(with)e(\(uniform\))d (mo)s(dulus)i Ft(\013)2807 6335 y Fu(f)2852 6320 y Fx(;)p eop %%Page: 8 8 8 7 bop 32 450 a Fx(\(b\))42 b(a)32 b(mo)s(dulus)g Ft(!)740 465 y Fu(f)796 450 y Fx(:)h Fn(N)43 b Fr(!)27 b Fn(N)48 b Fx(of)32 b(\(uniform\))e(con)m(tin)m(uit)m(y)-8 b(,)33 b(whic)m(h)g(satis\014es)860 624 y Fr(j)p Ft(a)22 b Fr(\000)h Ft(b)p Fr(j)28 b(\024)g Fx(2)1312 583 y Fv(\000)p Fu(!)1411 595 y Fk(f)1450 583 y Fm(\()p Fu(k)r Fm(\)+1)1666 624 y Fr(!)f(j)p Ft(h)1877 639 y Fu(f)1922 624 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)2329 639 y Fu(f)2374 624 y Fx(\()p Ft(b;)17 b(n)p Fx(\))p Fr(j)28 b(\024)g Fx(2)2803 583 y Fv(\000)p Fu(k)204 786 y Fx(for)k Ft(n)c Fr(\025)g Ft(\013)606 801 y Fu(f)651 786 y Fx(\()p Ft(k)s Fx(\).)44 b Ft(\013)914 801 y Fu(f)992 786 y Fx(and)32 b Ft(!)1242 801 y Fu(f)1320 786 y Fx(are)h(required)g(to)f(b)s(e)h(w)m (eakly)g(increasing.)137 926 y(Notice)j(that)g(a)g(con)m(tin)m(uous)h (function)e(is)h(giv)m(en)g(b)m(y)h(the)f(ob)5 b(jects)38 b Ft(h)2688 941 y Fu(f)2733 926 y Fx(,)f Ft(\013)2859 941 y Fu(f)2905 926 y Fx(,)g Ft(!)3030 941 y Fu(f)3111 926 y Fx(of)e(t)m(yp)s(e)i(lev)m(el)f(1)38 1042 y(only)-8 b(.)43 b(This)32 b(is)h(due)g(to)f(the)h(fact)f(that)h(it)e(su\016ces)k (to)d(de\014ne)i(its)e(v)-5 b(alues)33 b(on)f(rational)e(n)m(um)m(b)s (ers.)137 1158 y(An)j(example)f(is)g(the)h(squaring)g(function)f Fi(sq)11 b Fx(:)34 b([0)p Ft(;)17 b Fx(1])27 b Fr(!)g Fn(R)44 b Fx(giv)m(en)32 b(b)m(y)38 1298 y(\(a\))41 b(the)33 b(appro)m(ximating)d(map)i Ft(h)1295 1313 y Fl(sq)1363 1298 y Fx(\()p Ft(a;)17 b(n)p Fx(\))28 b(:=)f Ft(a)1801 1262 y Fm(2)1873 1298 y Fx(and)33 b(mo)s(dulus)e Ft(\013)2518 1313 y Fl(sq)2586 1298 y Fx(\()p Ft(k)s Fx(\))d(:=)f(0,)33 b(and)32 1414 y(\(b\))42 b(the)33 b(mo)s(dulus)e Ft(!)826 1429 y Fl(sq)894 1414 y Fx(\()p Ft(k)s Fx(\))c(:=)h Ft(k)d Fx(+)d(2)33 b(of)f(uniform)e(con)m(tin)m(uit)m(y)-8 b(,)33 b(for)f Fr(\000)p Ft(!)2681 1429 y Fl(sq)2749 1414 y Fx(\()p Ft(k)s Fx(\))22 b(+)g(1)28 b(=)f Fr(\000)p Ft(k)f Fr(\000)c Fx(1)33 b(and)773 1584 y Fr(j)p Ft(a)22 b Fr(\000)h Ft(b)p Fr(j)k(\024)i Fx(2)1225 1543 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1440 1584 y Fr(!)e(j)p Ft(a)1646 1543 y Fm(2)1708 1584 y Fr(\000)22 b Ft(b)1848 1543 y Fm(2)1888 1584 y Fr(j)27 b Fx(=)h Fr(j)p Fx(\()p Ft(a)22 b Fr(\000)h Ft(b)p Fx(\)\()p Ft(a)f Fx(+)g Ft(b)p Fx(\))p Fr(j)28 b(\024)g Fx(2)2863 1543 y Fv(\000)p Fu(k)2961 1584 y Ft(;)204 1747 y Fx(b)s(ecause)34 b Ft(a)22 b Fx(+)g Ft(b)28 b Fr(\024)h Fx(2.)38 1886 y(Similarly)h(all)i(p)s(olynomials)f(with)j (rational)e(co)s(e\016cien)m(ts)j(on)f(\014nite)g(in)m(terv)-5 b(als)33 b(can)i(b)s(e)f(view)m(ed)i(as)38 2002 y(con)m(tin)m(uous)d (functions)g(in)e(our)i(sense.)137 2119 y(Another)g(example)g(is)f(the) h(in)m(v)m(erse)h(function)e Fi(inv)12 b Fx(:)33 b([2)2139 2082 y Fv(\000)p Fu(l)2220 2119 y Ft(;)17 b(c)p Fx(])27 b Fr(!)h Fn(R)43 b Fx(giv)m(en)33 b(b)m(y)38 2258 y(\(a\))41 b(the)33 b(appro)m(ximating)d(map)i Ft(h)1295 2273 y Fl(inv)1386 2258 y Fx(\()p Ft(a;)17 b(n)p Fx(\))28 b(:=)1785 2219 y Fm(1)p 1783 2235 38 4 v 1783 2292 a Fu(a)1863 2258 y Fx(and)33 b(mo)s(dulus)f Ft(\013)2509 2273 y Fl(inv)2599 2258 y Fx(\()p Ft(k)s Fx(\))c(:=)g(0,)k(and)32 2375 y(\(b\))42 b(the)31 b(mo)s(dulus)f Ft(!)823 2390 y Fl(inv)914 2375 y Fx(\()p Ft(k)s Fx(\))e(:=)f Ft(k)22 b Fx(+)d(2)p Ft(l)i Fx(+)e(1)30 b(of)h(uniform)e(con)m(tin)m(uit)m(y)-8 b(,)32 b(for)e Fr(\000)p Ft(!)2881 2390 y Fl(inv)2972 2375 y Fx(\()p Ft(k)s Fx(\))19 b(+)g(1)27 b(=)h Fr(\000)p Ft(k)22 b Fr(\000)e Fx(2)p Ft(l)204 2492 y Fx(and)939 2642 y Fr(j)p Ft(a)i Fr(\000)h Ft(b)p Fr(j)28 b(\024)g Fx(2)1391 2601 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)p Fu(l)1628 2642 y Fr(!)1755 2528 y Fj(\014)1755 2588 y(\014)1755 2647 y(\014)1800 2575 y Fx(1)p 1798 2619 52 4 v 1798 2711 a Ft(a)1882 2642 y Fr(\000)1991 2575 y Fx(1)p 1991 2619 49 4 v 1995 2711 a Ft(b)2050 2528 y Fj(\014)2050 2588 y(\014)2050 2647 y(\014)2111 2642 y Fx(=)2214 2528 y Fj(\014)2214 2588 y(\014)2214 2647 y(\014)2258 2575 y Ft(b)22 b Fr(\000)h Ft(a)p 2258 2619 215 4 v 2318 2711 a(ab)2482 2528 y Fj(\014)2482 2588 y(\014)2482 2647 y(\014)2543 2642 y Fr(\024)28 b Fx(2)2697 2601 y Fv(\000)p Fu(k)2794 2642 y Ft(;)204 2827 y Fx(b)s(ecause)34 b Ft(ab)28 b Fr(\025)g Fx(2)839 2791 y Fv(\000)p Fm(2)p Fu(l)955 2827 y Fx(.)137 2967 y(Since)g(the)f(appro)m(ximating)e(map)h(op)s(erates)h (on)g(rationals)e(only)-8 b(,)28 b(w)m(e)g(need)g(to)e(de\014ne)i (separately)38 3083 y(what)33 b(it)e(means)i(to)f(apply)g(a)h(con)m (tin)m(uous)g(function)f(in)g(our)g(sense)j(to)d(a)g(real.)38 3223 y Fp(De\014nition)37 b(3.4.)43 b Fw(Applic)-5 b(ation)40 b Fx(of)34 b(a)f(con)m(tin)m(uous)i(function)e Ft(f)22 b Fx(:)33 b Ft(I)38 b Fr(!)29 b Fn(R)45 b Fx(\(giv)m(en)34 b(b)m(y)h Ft(h)3304 3238 y Fu(f)3349 3223 y Fx(,)f Ft(\013)3472 3238 y Fu(f)3518 3223 y Fx(,)g Ft(!)3640 3238 y Fu(f)3685 3223 y Fx(\))38 3339 y(to)e(a)g(real)g Ft(x)c Fx(:=)g(\(\()p Ft(a)769 3354 y Fu(n)816 3339 y Fx(\))854 3354 y Fu(n)901 3339 y Ft(;)17 b(M)10 b Fx(\))33 b(in)e Ft(I)41 b Fx(is)32 b(de\014ned)i(to)e(b)s(e)1630 3501 y(\()p Ft(h)1724 3516 y Fu(f)1770 3501 y Fx(\()p Ft(a)1859 3516 y Fu(n)1906 3501 y Ft(;)17 b(n)p Fx(\)\))2084 3516 y Fu(n)38 3664 y Fx(with)32 b(mo)s(dulus)f(max\()p Ft(\013)935 3679 y Fu(f)980 3664 y Fx(\()p Ft(k)26 b Fx(+)c(2\))p Ft(;)17 b(M)10 b Fx(\()p Ft(!)1527 3679 y Fu(f)1572 3664 y Fx(\()p Ft(k)25 b Fx(+)d(1\))g Fr(\000)g Fx(1\)\).)44 b(This)32 b(is)g(a)h(mo)s(dulus,)e(for)338 3822 y Fr(j)p Ft(h)422 3837 y Fu(f)467 3822 y Fx(\()p Ft(a)556 3837 y Fu(m)622 3822 y Ft(;)17 b(m)p Fx(\))23 b Fr(\000)f Ft(h)967 3837 y Fu(f)1013 3822 y Fx(\()p Ft(a)1102 3837 y Fu(n)1149 3822 y Ft(;)17 b(n)p Fx(\))p Fr(j)365 3973 y(\024)28 b(j)p Ft(h)554 3988 y Fu(f)600 3973 y Fx(\()p Ft(a)689 3988 y Fu(m)755 3973 y Ft(;)17 b(m)p Fx(\))p Fr(\000)q Ft(h)1056 3988 y Fu(f)1101 3973 y Fx(\()p Ft(a)1190 3988 y Fu(m)1257 3973 y Ft(;)g(p)p Fx(\))p Fr(j)k Fx(+)h Fr(j)p Ft(h)1619 3988 y Fu(f)1665 3973 y Fx(\()p Ft(a)1754 3988 y Fu(m)1820 3973 y Ft(;)17 b(p)p Fx(\))p Fr(\000)p Ft(h)2084 3988 y Fu(f)2130 3973 y Fx(\()p Ft(a)2219 3988 y Fu(n)2266 3973 y Ft(;)g(p)p Fx(\))p Fr(j)k Fx(+)h Fr(j)p Ft(h)2628 3988 y Fu(f)2674 3973 y Fx(\()p Ft(a)2763 3988 y Fu(n)2810 3973 y Ft(;)17 b(p)p Fx(\))p Fr(\000)p Ft(h)3074 3988 y Fu(f)3119 3973 y Fx(\()p Ft(a)3208 3988 y Fu(n)3256 3973 y Ft(;)g(n)p Fx(\))p Fr(j)365 4136 y(\024)28 b Fx(2)519 4095 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)729 4136 y Fx(+)22 b(2)876 4095 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1086 4136 y Fx(+)g(2)1233 4095 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)38 4305 y Fx(if)37 b Ft(m;)17 b(n)39 b Fr(\025)g Ft(M)10 b Fx(\()p Ft(!)678 4320 y Fu(f)724 4305 y Fx(\()p Ft(k)29 b Fx(+)e(1\))f Fr(\000)h Fx(1\))38 b(and)h Ft(p)f Fr(\025)h Ft(\013)1748 4320 y Fu(f)1794 4305 y Fx(\()p Ft(k)29 b Fx(+)d(1\))39 b(\(for)f(the)h(middle)e(term\),)j(and)f(moreo)m(v)m (er)38 4421 y Ft(m;)17 b(n;)g(p)29 b Fr(\025)g Ft(\013)515 4436 y Fu(f)560 4421 y Fx(\()p Ft(k)d Fx(+)c(2\))33 b(\(for)g(the)h (\014rst)f(and)h(last)e(term\).)45 b(W)-8 b(e)34 b(denote)g(this)f (real)f(b)m(y)i Ft(f)11 b Fx(\()p Ft(x)p Fx(\).)46 b(The)34 b(set)38 4537 y(of)e(all)e(suc)m(h)k(reals)f(is)f(called)f(the)i Fw(r)-5 b(ange)39 b Fx(of)33 b Ft(f)11 b Fx(.)137 4677 y(W)-8 b(e)33 b(w)m(an)m(t)h(to)e(sho)m(w)i(that)e(application)e(is)i (compatible)f(with)h(equalit)m(y)-8 b(.)38 4816 y Fp(Lemma)37 b(3.5.)42 b Fw(L)-5 b(et)35 b Ft(f)46 b Fw(b)-5 b(e)34 b(a)h(c)-5 b(ontinuous)34 b(function.)45 b(Then)1417 4979 y Ft(x)28 b Fx(=)g Ft(y)i Fr(!)e Ft(f)11 b Fx(\()p Ft(x)p Fx(\))27 b(=)h Ft(f)11 b Fx(\()p Ft(y)t Fx(\))p Ft(:)38 5152 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)30 b Ft(x)e Fx(:=)g(\(\()p Ft(a)849 5167 y Fu(n)896 5152 y Fx(\))934 5167 y Fu(n)981 5152 y Ft(;)17 b(M)10 b Fx(\))30 b(and)h Ft(y)f Fx(:=)e(\(\()p Ft(b)1711 5167 y Fu(n)1758 5152 y Fx(\))1796 5167 y Fu(n)1843 5152 y Ft(;)17 b(N)10 b Fx(\),)31 b(and)f(assume)h Ft(x)d Fx(=)f Ft(y)t Fx(.)42 b(W)-8 b(e)31 b(m)m(ust)f(sho)m(w)h(that)38 5268 y(\()p Ft(h)132 5283 y Fu(f)177 5268 y Fx(\()p Ft(a)266 5283 y Fu(n)313 5268 y Ft(;)17 b(n)p Fx(\)\))491 5283 y Fu(n)566 5268 y Fx(=)27 b(\()p Ft(h)763 5283 y Fu(f)809 5268 y Fx(\()p Ft(b)888 5283 y Fu(n)935 5268 y Ft(;)17 b(n)p Fx(\)\))1113 5283 y Fu(n)1160 5268 y Fx(.)43 b(This)33 b(follo)m(ws)e(from)329 5431 y Fr(j)p Ft(h)413 5446 y Fu(f)458 5431 y Fx(\()p Ft(a)547 5446 y Fu(n)594 5431 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)912 5446 y Fu(f)957 5431 y Fx(\()p Ft(b)1036 5446 y Fu(n)1084 5431 y Ft(;)17 b(n)p Fx(\))p Fr(j)356 5582 y(\024)29 b(j)p Ft(h)546 5597 y Fu(f)591 5582 y Fx(\()p Ft(a)680 5597 y Fu(n)727 5582 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)1044 5597 y Fu(f)1090 5582 y Fx(\()p Ft(a)1179 5597 y Fu(n)1226 5582 y Ft(;)17 b(p)p Fx(\))p Fr(j)22 b Fx(+)g Fr(j)p Ft(h)1589 5597 y Fu(f)1634 5582 y Fx(\()p Ft(a)1723 5597 y Fu(n)1770 5582 y Ft(;)17 b(p)p Fx(\))22 b Fr(\000)g Ft(h)2078 5597 y Fu(f)2124 5582 y Fx(\()p Ft(b)2203 5597 y Fu(n)2250 5582 y Ft(;)17 b(p)p Fx(\))p Fr(j)22 b Fx(+)g Fr(j)p Ft(h)2613 5597 y Fu(f)2658 5582 y Fx(\()p Ft(b)2737 5597 y Fu(n)2784 5582 y Ft(;)17 b(p)p Fx(\))22 b Fr(\000)g Ft(h)3092 5597 y Fu(f)3138 5582 y Fx(\()p Ft(b)3217 5597 y Fu(n)3264 5582 y Ft(;)17 b(n)p Fx(\))p Fr(j)356 5746 y(\024)29 b Fx(2)511 5705 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)720 5746 y Fx(+)22 b(2)867 5705 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1077 5746 y Fx(+)g(2)1224 5705 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)38 5914 y Fx(if)36 b Ft(n;)17 b(p)37 b Fr(\025)g Ft(\013)496 5929 y Fu(f)542 5914 y Fx(\()p Ft(k)28 b Fx(+)e(2\))37 b(\(the)i(\014rst)f(and)g(the)g(last)f(term\),) i(and)f(for)f(the)h(middle)e(term)i(in)f(addition)38 6030 y Ft(n)28 b Fr(\025)g Ft(p)k Fx(with)h Ft(p)f Fx(pro)m(vided)h(b)m (y)h(Lemma)d(1.3\(c\))h(\(for)g Ft(!)2023 6045 y Fu(f)2068 6030 y Fx(\()p Ft(k)25 b Fx(+)d(1\))g Fr(\000)h Fx(1\).)p 3682 5971 41 4 v 3682 6027 4 56 v 3720 6027 V 3682 6030 41 4 v 137 6204 a(Next)37 b(w)m(e)g(sho)m(w)g(that)e(indeed)h(a)g(con)m (tin)m(uous)g(function)f Ft(f)47 b Fx(has)36 b Ft(!)2591 6219 y Fu(f)2671 6204 y Fx(as)g(a)g(mo)s(dulus)e(of)i(uniform)38 6320 y(con)m(tin)m(uit)m(y)-8 b(.)p eop %%Page: 9 9 9 8 bop 38 450 a Fp(Lemma)37 b(3.6.)42 b Fw(L)-5 b(et)35 b Ft(f)46 b Fw(b)-5 b(e)34 b(c)-5 b(ontinuous)35 b(and)f Ft(x;)17 b(y)38 b Fw(b)-5 b(e)35 b(r)-5 b(e)g(als)34 b(in)h(its)g(domain.)43 b(Then)1023 625 y Fr(j)p Ft(x)22 b Fr(\000)g Ft(y)t Fr(j)27 b(\024)h Fx(2)1488 584 y Fv(\000)p Fu(!)1587 596 y Fk(f)1626 584 y Fm(\()p Fu(k)r Fm(\))1752 625 y Fr(!)f(j)p Ft(f)11 b Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(y)t Fx(\))p Fr(j)26 b(\024)j Fx(2)2614 584 y Fv(\000)p Fu(k)2711 625 y Ft(:)38 813 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)35 b Ft(x)e Fx(:=)f(\(\()p Ft(a)863 828 y Fu(n)910 813 y Fx(\))948 828 y Fu(n)995 813 y Ft(;)17 b(M)10 b Fx(\))36 b(and)f Ft(y)h Fx(:=)c(\(\()p Ft(b)1745 828 y Fu(n)1792 813 y Fx(\))1830 828 y Fu(n)1877 813 y Ft(;)17 b(N)10 b Fx(\).)52 b(Assume)36 b Fr(j)p Ft(a)2569 828 y Fu(n)2639 813 y Fr(\000)25 b Ft(b)2782 828 y Fu(n)2829 813 y Fr(j)32 b(\024)h Fx(2)3048 777 y Fv(\000)p Fu(!)3147 789 y Fk(f)3186 777 y Fm(\()p Fu(k)r Fm(\))3308 813 y Fx(+)24 b(2)3457 777 y Fv(\000)p Fu(l)3533 786 y Fh(0)3606 813 y Fx(for)38 930 y Ft(n)k Fr(\025)g Ft(p)278 945 y Fu(l)299 954 y Fh(0)338 930 y Fx(,)k(with)f Ft(l)647 945 y Fm(0)715 930 y Fx(:=)c Ft(!)906 945 y Fu(f)951 930 y Fx(\()p Ft(k)s Fx(\).)43 b(It)32 b(su\016ces)j(to)c(sho) m(w)i(that)f(for)f(an)h(arbitrary)f Ft(l)j Fx(w)m(e)f(can)g(\014nd)f Ft(q)3477 945 y Fu(l)3535 930 y Fx(suc)m(h)38 1048 y(that)g Fr(j)p Ft(h)333 1063 y Fu(f)378 1048 y Fx(\()p Ft(a)467 1063 y Fu(n)514 1048 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)832 1063 y Fu(f)878 1048 y Fx(\()p Ft(b)957 1063 y Fu(n)1004 1048 y Ft(;)17 b(n)p Fx(\))p Fr(j)27 b(\024)h Fx(2)1353 1012 y Fv(\000)p Fu(k)1473 1048 y Fx(+)22 b(2)1620 1012 y Fv(\000)p Fu(l)1733 1048 y Fx(for)32 b Ft(n)c Fr(\025)g Ft(q)2116 1063 y Fu(l)2142 1048 y Fx(.)44 b(This)33 b(follo)m(ws)e(from)329 1221 y Fr(j)p Ft(h)413 1236 y Fu(f)458 1221 y Fx(\()p Ft(a)547 1236 y Fu(n)594 1221 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)912 1236 y Fu(f)957 1221 y Fx(\()p Ft(b)1036 1236 y Fu(n)1084 1221 y Ft(;)17 b(n)p Fx(\))p Fr(j)356 1372 y(\024)29 b(j)p Ft(h)546 1387 y Fu(f)591 1372 y Fx(\()p Ft(a)680 1387 y Fu(n)727 1372 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)1044 1387 y Fu(f)1090 1372 y Fx(\()p Ft(a)1179 1387 y Fu(n)1226 1372 y Ft(;)17 b(p)p Fx(\))p Fr(j)22 b Fx(+)g Fr(j)p Ft(h)1589 1387 y Fu(f)1634 1372 y Fx(\()p Ft(a)1723 1387 y Fu(n)1770 1372 y Ft(;)17 b(p)p Fx(\))22 b Fr(\000)g Ft(h)2078 1387 y Fu(f)2124 1372 y Fx(\()p Ft(b)2203 1387 y Fu(n)2250 1372 y Ft(;)17 b(p)p Fx(\))p Fr(j)22 b Fx(+)g Fr(j)p Ft(h)2613 1387 y Fu(f)2658 1372 y Fx(\()p Ft(b)2737 1387 y Fu(n)2784 1372 y Ft(;)17 b(p)p Fx(\))22 b Fr(\000)g Ft(h)3092 1387 y Fu(f)3138 1372 y Fx(\()p Ft(b)3217 1387 y Fu(n)3264 1372 y Ft(;)17 b(n)p Fx(\))p Fr(j)356 1535 y(\024)29 b Fx(2)511 1494 y Fv(\000)p Fu(l)q Fv(\000)p Fm(1)703 1535 y Fx(+)23 b(2)851 1494 y Fv(\000)p Fu(k)970 1535 y Fx(+)f(2)1117 1494 y Fv(\000)p Fu(l)q Fv(\000)p Fm(1)38 1714 y Fx(if)38 b Ft(n;)17 b(p)38 b Fr(\025)h Ft(\013)501 1729 y Fu(f)547 1714 y Fx(\()p Ft(l)29 b Fx(+)d(1\))39 b(\(the)g(\014rst)h(and)f(the)g(last)f(term\),)j(and)e (for)f(the)i(middle)d(term)h(in)h(addition)38 1836 y Ft(p)49 b Fr(\025)i Ft(\013)326 1851 y Fu(f)371 1836 y Fx(\()p Ft(k)s Fx(\),)e(b)s(ecause)e(then)f(the)g(assumption)f Fr(j)p Ft(a)1975 1851 y Fu(n)2052 1836 y Fr(\000)32 b Ft(b)2202 1851 y Fu(n)2249 1836 y Fr(j)50 b(\024)g Fx(2)2503 1800 y Fv(\000)p Fu(!)2602 1812 y Fk(f)2641 1800 y Fm(\()p Fu(k)r Fm(\)+1)2874 1836 y Fx(implies)43 b Fr(j)p Ft(h)3302 1851 y Fu(f)3348 1836 y Fx(\()p Ft(a)3437 1851 y Fu(n)3484 1836 y Ft(;)17 b(p)p Fx(\))30 b Fr(\000)38 1955 y Ft(h)94 1970 y Fu(f)139 1955 y Fx(\()p Ft(b)218 1970 y Fu(n)266 1955 y Ft(;)17 b(p)p Fx(\))p Fr(j)27 b(\024)h Fx(2)606 1918 y Fv(\000)p Fu(k)703 1955 y Fx(.)p 3682 1895 41 4 v 3682 1951 4 56 v 3720 1951 V 3682 1954 41 4 v 137 2140 a(W)-8 b(e)33 b(de\014ne)h(the)f(comp)s(osition)d(of)i(t)m(w)m(o)i (con)m(tin)m(uous)f(functions.)38 2284 y Fp(De\014nition)53 b(3.7.)c Fx(Assume)f(that)f Ft(f)21 b Fx(:)39 b Ft(I)60 b Fr(!)52 b Fn(R)58 b Fx(and)47 b Ft(g)15 b Fx(:)38 b Ft(J)61 b Fr(!)52 b Fn(R)58 b Fx(are)47 b(con)m(tin)m(uous,)52 b(and)47 b(that)38 2401 y Ft(h)94 2416 y Fu(f)139 2401 y Fx([\()p Ft(I)30 b Fr(\\)23 b Fn(Q)11 b Fx(\))28 b Fr(\002)23 b Fn(N)9 b Fx(])34 b Fr(\022)28 b Ft(J)9 b Fx(.)44 b(Then)33 b(the)g(function)f Ft(g)26 b Fr(\016)c Ft(f)g Fx(:)33 b Ft(I)i Fr(!)28 b Fn(R)43 b Fx(is)32 b(de\014ned)i(b)m(y)38 2545 y(\(a\))41 b(the)25 b(appro)m(ximating)e (map)h Ft(h)11 b Fx(:)33 b(\()p Ft(I)14 b Fr(\\)6 b Fn(Q)13 b Fx(\))6 b Fr(\002)h Fn(N)48 b Fr(!)28 b Fn(Q)11 b Fx(,)33 b Ft(h)p Fx(\()p Ft(a;)17 b(n)p Fx(\))27 b(:=)h Ft(h)2574 2560 y Fu(g)2614 2545 y Fx(\()p Ft(h)2708 2560 y Fu(f)2754 2545 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(n)p Fx(\))24 b(with)h(mo)s(dulus)204 2666 y Ft(\013)266 2681 y Fu(g)r Fv(\016)p Fu(f)383 2666 y Fx(\()p Ft(k)s Fx(\))i(:=)h(max)852 2585 y Fj(\000)898 2666 y Ft(\013)960 2681 y Fu(g)1000 2666 y Fx(\()p Ft(k)d Fx(+)d(2\))p Ft(;)17 b(\013)1405 2681 y Fu(f)1450 2666 y Fx(\()p Ft(!)1549 2681 y Fu(g)1589 2666 y Fx(\()p Ft(k)25 b Fx(+)d(1\))g Fr(\000)h Fx(1\))2097 2585 y Fj(\001)2142 2666 y Fx(;)32 2784 y(\(b\))42 b(the)33 b(mo)s(dulus)e Ft(!)826 2799 y Fu(g)r Fv(\016)p Fu(f)943 2784 y Fx(\()p Ft(k)s Fx(\))c(:=)h Ft(!)1292 2799 y Fu(f)1337 2784 y Fx(\()p Ft(!)1436 2799 y Fu(g)1476 2784 y Fx(\()p Ft(k)s Fx(\))22 b Fr(\000)h Fx(1\))f(+)g(1)32 b(of)g(\(uniform\))f(con) m(tin)m(uit)m(y)h(for)g Ft(g)26 b Fr(\016)c Ft(f)11 b Fx(.)137 2928 y(W)-8 b(e)33 b(need)h(to)e(sho)m(w)i(that)e(this)h (indeed)f(de\014nes)j(a)d(con)m(tin)m(uous)h(function.)38 3073 y Fp(Lemma)k(3.8.)42 b Fw(Under)35 b(the)f(assumptions)g(of)h(the) g(de\014nition)f(ab)-5 b(ove)34 b(we)g(have)38 3217 y Fx(\(a\))41 b Ft(\013)266 3232 y Fu(g)r Fv(\016)p Fu(f)417 3217 y Fw(is)35 b(a)42 b Fs(Ca)n(uchy)36 b Fw(mo)-5 b(dulus;)32 3333 y Fx(\(b\))42 b Ft(!)265 3348 y Fu(g)r Fv(\016)p Fu(f)416 3333 y Fw(is)35 b(a)f(mo)-5 b(dulus)35 b(of)g(uniform)f(c)-5 b(ontinuity)35 b(for)g Ft(g)25 b Fr(\016)d Ft(f)11 b Fw(.)38 3519 y(Pr)-5 b(o)g(of.)41 b Fx(\(a\).)304 3683 y Fr(j)p Ft(h)388 3698 y Fu(g)428 3683 y Fx(\()p Ft(h)522 3698 y Fu(f)568 3683 y Fx(\()p Ft(a;)17 b(m)p Fx(\))p Ft(;)g(m)p Fx(\))22 b Fr(\000)h Ft(h)1169 3698 y Fu(g)1209 3683 y Fx(\()p Ft(h)1303 3698 y Fu(f)1348 3683 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(n)p Fx(\))p Fr(j)28 b(\024)g(j)p Ft(h)1962 3698 y Fu(g)2002 3683 y Fx(\()p Ft(h)2096 3698 y Fu(f)2141 3683 y Fx(\()p Ft(a;)17 b(m)p Fx(\))p Ft(;)g(m)p Fx(\))22 b Fr(\000)h Ft(h)2742 3698 y Fu(g)2782 3683 y Fx(\()p Ft(h)2876 3698 y Fu(f)2922 3683 y Fx(\()p Ft(a;)17 b(m)p Fx(\))p Ft(;)g(p)p Fx(\))p Fr(j)k Fx(+)1878 3834 y Fr(j)p Ft(h)1962 3849 y Fu(g)2002 3834 y Fx(\()p Ft(h)2096 3849 y Fu(f)2141 3834 y Fx(\()p Ft(a;)c(m)p Fx(\))p Ft(;)g(p)p Fx(\))22 b Fr(\000)h Ft(h)2706 3849 y Fu(g)2746 3834 y Fx(\()p Ft(h)2840 3849 y Fu(f)2885 3834 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(p)p Fx(\))p Fr(j)22 b Fx(+)1878 3985 y Fr(j)p Ft(h)1962 4000 y Fu(g)2002 3985 y Fx(\()p Ft(h)2096 4000 y Fu(f)2141 3985 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(p)p Fx(\))22 b Fr(\000)h Ft(h)2679 4000 y Fu(g)2719 3985 y Fx(\()p Ft(h)2813 4000 y Fu(f)2858 3985 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(n)p Fx(\))p Fr(j)1773 4149 y(\024)28 b Fx(2)1927 4108 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)2136 4149 y Fx(+)22 b(2)2283 4108 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2493 4149 y Fx(+)g(2)2640 4108 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)38 4327 y Fx(if)38 b Ft(m;)17 b(n)39 b Fr(\025)h Ft(\013)539 4342 y Fu(f)584 4327 y Fx(\()p Ft(!)683 4342 y Fu(g)723 4327 y Fx(\()p Ft(k)30 b Fx(+)c(1\))h Fr(\000)g Fx(1\))39 b(and)g Ft(p)g Fr(\025)h Ft(\013)1751 4342 y Fu(g)1791 4327 y Fx(\()p Ft(k)29 b Fx(+)e(1\))39 b(\(for)f(the)i(middle)e(term\),)i(and)f(moreo) m(v)m(er)38 4444 y Ft(m;)17 b(n;)g(p)27 b Fr(\025)i Ft(\013)513 4459 y Fu(g)553 4444 y Fx(\()p Ft(k)c Fx(+)d(2\))32 b(\(for)g(the)h (\014rst)g(and)g(last)f(term\).)137 4566 y(\(b\).)43 b(Assume)30 b Fr(j)p Ft(a)16 b Fr(\000)g Ft(b)p Fr(j)27 b(\024)h Fx(2)1133 4529 y Fv(\000)p Fu(!)1232 4541 y Fk(f)1272 4529 y Fm(\()p Fu(!)1343 4537 y Fk(g)1379 4529 y Fm(\()p Fu(k)r Fm(\))p Fv(\000)p Fm(1\)+1)1685 4566 y Fx(.)42 b(Then)31 b Fr(j)p Ft(h)2090 4581 y Fu(f)2135 4566 y Fx(\()p Ft(a;)17 b(n)p Fx(\))f Fr(\000)g Ft(h)2529 4581 y Fu(f)2574 4566 y Fx(\()p Ft(b;)h(n)p Fx(\))p Fr(j)27 b(\024)i Fx(2)3003 4529 y Fv(\000)p Fu(!)3102 4537 y Fk(g)3137 4529 y Fm(\()p Fu(k)r Fm(\)+1)3354 4566 y Fx(pro)m(vided)38 4684 y Ft(n)f Fr(\025)g Ft(\013)291 4699 y Fu(f)336 4684 y Fx(\()p Ft(!)435 4699 y Fu(g)475 4684 y Fx(\()p Ft(k)s Fx(\))22 b Fr(\000)g Fx(1\),)33 b(and)f(therefore)h Fr(j)p Ft(h)1555 4699 y Fu(g)1595 4684 y Fx(\()p Ft(h)1689 4699 y Fu(f)1735 4684 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)g(n)p Fx(\))k Fr(\000)i Ft(h)2281 4699 y Fu(g)2321 4684 y Fx(\()p Ft(h)2415 4699 y Fu(f)2460 4684 y Fx(\()p Ft(b;)17 b(n)p Fx(\))p Ft(;)g(n)p Fx(\))p Fr(j)28 b(\024)g Fx(2)3029 4648 y Fv(\000)p Fu(k)3159 4684 y Fx(pro)m(vided)33 b(also)38 4800 y Ft(n)c Fr(\025)g Ft(\013)293 4815 y Fu(g)333 4800 y Fx(\()p Ft(k)s Fx(\).)45 b(Both)34 b(conditions)e(on)h Ft(n)g Fx(hold)g(for)f Ft(n)d Fr(\025)h Ft(\013)2096 4815 y Fu(g)r Fv(\016)p Fu(f)2212 4800 y Fx(\()p Ft(k)s Fx(\),)k(since)f Ft(\013)2704 4815 y Fu(g)2744 4800 y Fx(,)h Ft(\013)2867 4815 y Fu(f)2945 4800 y Fx(and)g Ft(!)3197 4815 y Fu(g)3270 4800 y Fx(are)f(w)m(eakly)38 4917 y(increasing.)p 3682 4857 V 3682 4913 4 56 v 3720 4913 V 3682 4916 41 4 v 137 5102 a(F)-8 b(or)49 b(example,)54 b(assume)c(that)f Ft(f)22 b Fx(:)39 b Ft(I)65 b Fr(!)56 b Fn(R)61 b Fx(is)49 b(con)m(tin)m(uous)h(and)g(that)f(for)g(some)h Ft(l)r Fx(,)k(2)3508 5066 y Fv(\000)p Fu(l)3645 5102 y Fr(\024)38 5218 y Ft(h)94 5233 y Fu(f)139 5218 y Fx(\()p Ft(c;)17 b(n)p Fx(\))49 b Fr(2)h Fn(Q)63 b Fx(for)45 b(all)e Ft(n)i Fx(and)h(all)d(rationals)g Ft(c)50 b Fr(2)f Ft(I)8 b Fx(.)82 b(Then)46 b(w)m(e)h(can)e(form)f(the)i(in)m(v)m(erse)h (1)p Ft(=f)38 5334 y Fx(as)33 b(a)f(con)m(tin)m(uous)h(map)f Ft(I)j Fr(!)28 b Fn(R)5 b Fx(,)38 b(b)m(y)c(comp)s(osing)d Fi(inv)23 b Fr(\016)f Ft(f)11 b Fx(.)137 5451 y(Since)48 b(ev)m(ery)i(con)m(tin)m(uous)e(function)f(on)h(a)f(compact)g(in)m (terv)-5 b(al)46 b(comes)i(with)f(a)h(mo)s(dulus)e(of)38 5567 y(uniform)40 b(con)m(tin)m(uit)m(y)-8 b(,)45 b(w)m(e)e(can)g (explicitely)e(construct)i(a)f(b)s(ound)h(for)e(its)h(v)-5 b(alues,)45 b(ev)m(en)f(for)d(the)38 5683 y(v)-5 b(alues)32 b(of)g(its)g(appro)m(ximating)f(map.)38 5827 y Fp(Lemma)43 b(3.9.)i Fw(L)-5 b(et)41 b Ft(f)22 b Fx(:)35 b([)p Ft(a;)17 b(b)p Fx(])38 b Fr(!)f Fn(R)51 b Fw(b)-5 b(e)40 b(c)-5 b(ontinuous,)41 b(given)e(by)h Ft(h)2542 5842 y Fu(f)2588 5827 y Fw(,)h Ft(\013)2721 5842 y Fu(f)2807 5827 y Fw(and)e Ft(!)3062 5842 y Fu(f)3107 5827 y Fw(.)61 b(Then)39 b(for)h(al)5 b(l)38 5944 y Ft(n)28 b Fr(\025)g Ft(n)287 5959 y Fm(0)354 5944 y Fx(:=)g Ft(\013)547 5959 y Fu(f)592 5944 y Fx(\(0\))35 b Fw(and)f(r)-5 b(ationals)34 b Ft(c)28 b Fr(2)g Ft(I)1079 6116 y Fr(j)p Ft(h)1163 6131 y Fu(f)1208 6116 y Fx(\()p Ft(c;)17 b(n)p Fx(\))p Fr(j)27 b(\024)i Ft(M)38 b Fx(:=)28 b Fr(j)p Ft(h)1936 6131 y Fu(f)1981 6116 y Fx(\()p Ft(a;)17 b(n)2172 6131 y Fm(0)2211 6116 y Fx(\))p Fr(j)22 b Fx(+)g Ft(N)33 b Fx(+)22 b(1)p Ft(;)38 6295 y Fw(wher)-5 b(e)34 b Fx(\()p Ft(c)22 b Fr(\000)h Ft(a)p Fx(\)2)653 6259 y Fu(!)697 6271 y Fk(f)736 6259 y Fm(\(0\))p Fv(\000)p Fm(1)948 6295 y Fr(\024)29 b Ft(N)38 b Fr(2)28 b Fn(N)9 b Fw(.)51 b(Henc)-5 b(e)34 b(the)h(r)-5 b(ange)34 b(of)h Ft(f)45 b Fw(is)35 b(b)-5 b(ounde)g(d)34 b(ab)-5 b(ove)34 b(by)h Ft(M)10 b Fw(.)p eop %%Page: 10 10 10 9 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(Recall)31 b(that)h(for)g Ft(c;)17 b(d)27 b Fr(2)h Fx([)p Ft(a;)17 b(b)p Fx(])858 644 y Fr(j)p Ft(c)k Fr(\000)i Ft(d)p Fr(j)k(\024)h Fx(2)1309 603 y Fv(\000)p Fu(!)1408 615 y Fk(f)1448 603 y Fm(\(0\)+1)1660 644 y Fr(!)f(j)p Ft(h)1871 659 y Fu(f)1916 644 y Fx(\()p Ft(c;)17 b(n)2098 659 y Fm(0)2137 644 y Fx(\))23 b Fr(\000)f Ft(h)2353 659 y Fu(f)2399 644 y Fx(\()p Ft(d;)17 b(n)2590 659 y Fm(0)2629 644 y Fx(\))p Fr(j)27 b(\024)h Fx(1)p Ft(:)38 830 y Fx(Let)33 b Ft(")27 b Fx(:=)h(\()p Ft(c)22 b Fr(\000)g Ft(a)p Fx(\))p Ft(=)-5 b(N)43 b Fx(and)32 b Ft(a)1112 845 y Fu(i)1168 830 y Fx(:=)c Ft(a)22 b Fx(+)g Ft(i")33 b Fx(for)f Ft(i)c Fx(=)f(0)p Ft(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fx(.)44 b(Then)33 b Ft(a)2627 845 y Fu(N)2723 830 y Fx(=)27 b Ft(a)22 b Fx(+)g Ft(N)10 b(")28 b Fx(=)g Ft(c)k Fx(and)619 1028 y Fj(\014)619 1088 y(\014)652 1113 y Ft(h)708 1128 y Fu(f)754 1113 y Fx(\()p Ft(c;)17 b(n)936 1128 y Fm(0)975 1113 y Fx(\))22 b Fr(\000)h Ft(h)1191 1128 y Fu(f)1236 1113 y Fx(\()p Ft(a;)17 b(n)1427 1128 y Fm(0)1467 1113 y Fx(\))1505 1028 y Fj(\014)1505 1088 y(\014)1566 1113 y Fr(\024)1671 989 y Fu(N)7 b Fv(\000)p Fm(1)1675 1018 y Fj(X)1690 1228 y Fu(i)p Fm(=0)1824 1028 y Fj(\014)1824 1088 y(\014)1857 1113 y Ft(h)1913 1128 y Fu(f)1959 1113 y Fx(\()p Ft(a)2048 1128 y Fu(i)p Fm(+1)2166 1113 y Ft(;)17 b(n)2268 1128 y Fm(0)2308 1113 y Fx(\))22 b Fr(\000)g Ft(h)2523 1128 y Fu(f)2569 1113 y Fx(\()p Ft(a)2658 1128 y Fu(i)2686 1113 y Ft(;)17 b(n)2788 1128 y Fm(0)2827 1113 y Fx(\))2865 1028 y Fj(\014)2865 1088 y(\014)2926 1113 y Fr(\024)28 b Ft(N)5 b(;)38 1391 y Fx(hence)42 b Fr(j)p Ft(h)401 1406 y Fu(f)447 1391 y Fx(\()p Ft(c;)17 b(n)629 1406 y Fm(0)668 1391 y Fx(\))p Fr(j)42 b(\024)h(j)p Ft(h)980 1406 y Fu(f)1025 1391 y Fx(\()p Ft(a;)17 b(n)1216 1406 y Fm(0)1256 1391 y Fx(\))p Fr(j)27 b Fx(+)h Ft(N)52 b Fx(and)41 b(therefore)h Fr(j)p Ft(h)2283 1406 y Fu(f)2329 1391 y Fx(\()p Ft(c;)17 b(n)p Fx(\))p Fr(j)42 b(\024)g(j)p Ft(h)2822 1406 y Fu(f)2868 1391 y Fx(\()p Ft(a;)17 b(n)3059 1406 y Fm(0)3098 1391 y Fx(\))p Fr(j)28 b Fx(+)g Ft(N)38 b Fx(+)28 b(1)41 b(for)38 1507 y Ft(n)28 b Fr(\025)g Ft(n)287 1522 y Fm(0)354 1507 y Fx(:=)g Ft(\013)547 1522 y Fu(f)592 1507 y Fx(\(0\).)p 3682 1448 41 4 v 3682 1504 4 56 v 3720 1504 V 3682 1507 41 4 v 137 1709 a(The)39 b(suprem)m(um)e(of)g(the)g(range)g(of)g(a)g(con)m(tin)m(uous)h (function)e(on)i(a)e(compact)h(in)m(terv)-5 b(al)36 b(can)i(b)s(e)38 1825 y(sho)m(wn)e(to)e(exist)h(constructiv)m(ely)-8 b(.)1302 1787 y Fm(1)1387 1825 y Fx(W)g(e)35 b(pro)m(v)m(e)h(that)e(the)h(range) g(is)f(order)h(lo)s(cated)e(ab)s(o)m(v)m(e,)j(whic)m(h)38 1942 y(en)m(tails)c(\(b)m(y)h(Theorem)g(3.2\))f(that)g(it)g(has)h(a)f (suprem)m(um.)38 2093 y Fp(Lemma)37 b(3.10.)42 b Fw(L)-5 b(et)35 b Ft(f)22 b Fx(:)33 b Ft(I)i Fr(!)28 b Fn(R)46 b Fw(b)-5 b(e)34 b(c)-5 b(ontinuous,)34 b Ft(I)43 b Fw(c)-5 b(omp)g(act)34 b(with)h(r)-5 b(ational)34 b(end)g(p)-5 b(oints.)45 b(Then)38 2209 y(the)35 b(r)-5 b(ange)34 b(of)h Ft(f)45 b Fw(is)35 b(or)-5 b(der)34 b(lo)-5 b(c)g(ate)g(d)35 b(ab)-5 b(ove.)38 2411 y(Pr)g(o)g(of.)41 b Fx(Let)g Ft(h)575 2426 y Fu(f)621 2411 y Fx(,)j Ft(\013)754 2426 y Fu(f)840 2411 y Fx(and)e Ft(!)1100 2426 y Fu(f)1186 2411 y Fx(b)s(e)g(the)f (data)g(for)g Ft(f)11 b Fx(.)70 b(Giv)m(en)41 b Ft(a;)17 b(b)42 b Fx(with)f Ft(a)h(<)h(b)p Fx(,)h(\014x)e Ft(k)i Fx(suc)m(h)f(that)38 2533 y(2)87 2497 y Fv(\000)p Fu(k)220 2533 y Fr(\024)344 2494 y Fm(1)p 344 2510 36 4 v 344 2567 a(3)389 2533 y Fx(\()p Ft(b)26 b Fr(\000)g Ft(a)p Fx(\).)59 b(T)-8 b(ak)m(e)39 b(a)e(partition)e Ft(a)1561 2548 y Fm(0)1601 2533 y Ft(;)17 b(:)g(:)g(:)f(;)h(a)1871 2548 y Fu(l)1934 2533 y Fx(of)38 b Ft(I)45 b Fx(of)37 b(mesh)h Fr(\024)f Fx(2)2673 2497 y Fv(\000)p Fu(!)2772 2509 y Fk(f)2811 2497 y Fm(\()p Fu(k)r Fm(\)+2)2999 2533 y Fx(.)58 b(Then)39 b(for)e(ev)m(ery)38 2661 y Ft(c)k Fr(2)h Ft(I)48 b Fx(there)41 b(is)g(an)f Ft(i)h Fx(suc)m(h)h(that)e Fr(j)p Ft(c)27 b Fr(\000)i Ft(a)1602 2676 y Fu(i)1630 2661 y Fr(j)41 b(\024)h Fx(2)1867 2625 y Fv(\000)p Fu(!)1966 2637 y Fk(f)2005 2625 y Fm(\()p Fu(k)r Fm(\)+1)2193 2661 y Fx(.)67 b(Let)41 b Ft(n)2528 2676 y Fu(k)2612 2661 y Fx(:=)g Ft(\013)2818 2676 y Fu(f)2864 2661 y Fx(\()p Ft(k)s Fx(\))f(and)h(consider)g(all)38 2778 y(\014nitely)32 b(man)m(y)1334 2923 y Ft(h)p Fx(\()p Ft(a)1479 2938 y Fu(i)1507 2923 y Ft(;)17 b(n)1609 2938 y Fu(k)1652 2923 y Fx(\))98 b(for)32 b Ft(i)c Fx(=)f(0)p Ft(;)17 b(:)g(:)g(:)f(;)h(l)r Fx(.)38 3089 y(Let)33 b Ft(h)p Fx(\()p Ft(a)358 3104 y Fu(j)394 3089 y Ft(;)17 b(n)496 3104 y Fu(k)539 3089 y Fx(\))32 b(b)s(e)h(the)g(maxim)m(um)d(of)j(all)d(those.)137 3207 y Fw(Case)40 b Ft(h)p Fx(\()p Ft(a)527 3222 y Fu(j)564 3207 y Ft(;)17 b(n)666 3222 y Fu(k)708 3207 y Fx(\))28 b Fr(\024)h Ft(a)22 b Fx(+)1061 3168 y Fm(1)p 1061 3184 V 1061 3241 a(3)1106 3207 y Fx(\()p Ft(b)h Fr(\000)g Ft(a)p Fx(\).)44 b(W)-8 b(e)33 b(sho)m(w)h(that)e Ft(f)11 b Fx(\()p Ft(x)p Fx(\))28 b Fr(\024)h Ft(b)k Fx(for)f(all)f Ft(x)p Fx(.)44 b(Let)33 b Ft(x)28 b Fx(=)g(\(\()p Ft(b)3377 3222 y Fu(n)3424 3207 y Fx(\))3462 3222 y Fu(n)3509 3207 y Ft(;)17 b(M)10 b Fx(\).)38 3323 y(Then)34 b(for)e Ft(n)27 b Fr(\025)i Ft(n)691 3338 y Fu(k)453 3504 y Ft(h)509 3519 y Fu(f)554 3504 y Fx(\()p Ft(b)633 3519 y Fu(n)681 3504 y Ft(;)17 b(n)p Fx(\))27 b Fr(\024)h Ft(h)1009 3519 y Fu(f)1055 3504 y Fx(\()p Ft(b)1134 3519 y Fu(n)1181 3504 y Ft(;)17 b(n)1283 3519 y Fu(k)1326 3504 y Fx(\))22 b(+)g(2)1533 3462 y Fv(\000)p Fu(k)848 3667 y Fr(\024)28 b Ft(h)1009 3682 y Fu(f)1055 3667 y Fx(\()p Ft(a)1144 3682 y Fu(i)1172 3667 y Ft(;)17 b(n)1274 3682 y Fu(k)1317 3667 y Fx(\))22 b(+)g(2)1524 3626 y Fv(\000)p Fu(k)r Fm(+1)1809 3667 y Fx(for)32 b Ft(i)h Fx(suc)m(h)h(that)e Fr(j)p Ft(b)2524 3682 y Fu(n)2593 3667 y Fr(\000)23 b Ft(a)2744 3682 y Fu(i)2772 3667 y Fr(j)28 b(\024)g Fx(2)2982 3631 y Fv(\000)p Fu(!)3081 3643 y Fk(f)3120 3631 y Fm(\()p Fu(k)r Fm(\)+1)848 3831 y Fr(\024)g Ft(h)1009 3846 y Fu(f)1055 3831 y Fx(\()p Ft(a)1144 3846 y Fu(j)1180 3831 y Ft(;)17 b(n)1282 3846 y Fu(k)1325 3831 y Fx(\))22 b(+)g(2)1532 3790 y Fv(\000)p Fu(k)r Fm(+1)848 3976 y Fr(\024)28 b Ft(b:)137 4176 y Fw(Case)39 b Ft(a)21 b Fx(+)560 4136 y Fm(1)p 560 4153 V 560 4210 a(3)605 4176 y Fx(\()p Ft(b)g Fr(\000)g Ft(a)p Fx(\))28 b Ft(<)g(h)p Fx(\()p Ft(a)1169 4191 y Fu(j)1205 4176 y Ft(;)17 b(n)1307 4191 y Fu(k)1350 4176 y Fx(\).)43 b(W)-8 b(e)32 b(sho)m(w)h Ft(a)28 b Fr(\024)h Ft(f)11 b Fx(\()p Ft(x)p Fx(\))31 b(for)h Ft(x)c Fx(:=)g Ft(a)2686 4191 y Fu(j)2722 4176 y Fx(.)43 b(Then)34 b Ft(f)11 b Fx(\()p Ft(x)p Fx(\))31 b(is)h(giv)m(en)g(b)m(y)38 4293 y(the)h Fs(Ca)n(uchy)i Fx(sequence)g(\()p Ft(h)1118 4308 y Fu(f)1163 4293 y Fx(\()p Ft(a)1252 4308 y Fu(j)1289 4293 y Ft(;)17 b(n)p Fx(\)\))1467 4308 y Fu(n)1514 4293 y Fx(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(for)e Ft(n)c Fr(\025)g Ft(n)2375 4308 y Fu(k)720 4531 y Ft(h)776 4546 y Fu(f)822 4531 y Fx(\()p Ft(a)911 4546 y Fu(j)947 4531 y Ft(;)17 b(n)p Fx(\))28 b Fr(\025)g Ft(h)1276 4546 y Fu(f)1322 4531 y Fx(\()p Ft(a)1411 4546 y Fu(j)1447 4531 y Ft(;)17 b(n)1549 4546 y Fu(k)1592 4531 y Fx(\))22 b Fr(\000)h Fx(2)1801 4489 y Fv(\000)p Fu(k)1926 4531 y Fr(\025)28 b Ft(a)22 b Fx(+)2212 4463 y(1)p 2212 4508 49 4 v 2212 4599 a(3)2271 4531 y(\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))f Fr(\000)h Fx(2)2732 4489 y Fv(\000)p Fu(k)2857 4531 y Fr(\025)28 b Ft(a:)38 4752 y Fx(Hence)34 b Ft(a)28 b Fr(\024)g Ft(f)11 b Fx(\()p Ft(x)p Fx(\).)p 3682 4693 41 4 v 3682 4748 4 56 v 3720 4748 V 3682 4751 41 4 v 38 4953 a Fp(Corollary)50 b(3.11.)e Fw(L)-5 b(et)46 b Ft(f)22 b Fx(:)37 b Ft(I)55 b Fr(!)47 b Fn(R)57 b Fw(b)-5 b(e)45 b(c)-5 b(ontinuous,)48 b Ft(I)53 b Fw(c)-5 b(omp)g(act)45 b(with)g(r)-5 b(ational)45 b(end)g(p)-5 b(oints.)38 5070 y(Then)34 b(the)h(r)-5 b(ange)34 b(of)h Ft(f)45 b Fw(has)35 b(a)f(supr)-5 b(emum,)35 b(denote)-5 b(d)34 b Fr(j)-17 b(j)p Ft(f)11 b Fr(j)-17 b(j)2179 5085 y Fu(I)2218 5070 y Fw(.)38 5271 y(Pr)-5 b(o)g(of.)41 b Fx(By)23 b(Lemma)d(3.9)i(w)m(e)i (kno)m(w)f(that)f(the)g(range)h(of)e Ft(f)33 b Fx(is)22 b(b)s(ounded)h(ab)s(o)m(v)m(e,)i(and)d(b)m(y)i(Lemma)c(3.10)38 5388 y(it)31 b(is)h(order)h(lo)s(cated)f(ab)s(o)m(v)m(e.)44 b(Hence)34 b(b)m(y)g(Theorem)e(3.2)h(it)e(has)i(a)f(suprem)m(um.)p 3682 5329 V 3682 5384 4 56 v 3720 5384 V 3682 5387 41 4 v 137 5589 a(W)-8 b(e)27 b(next)g(supply)g(the)g(standard)f (constructiv)m(e)i(v)m(ersions)f(of)f(the)h Fw(interme)-5 b(diate)28 b(value)h(the)-5 b(or)g(em)7 b Fx(.)38 5741 y Fp(Theorem)50 b(3.12)h Fx(\(Appro)m(ximate)43 b(In)m(termediate)h(V) -8 b(alue)44 b(Theorem\))p Fp(.)k Fw(L)-5 b(et)46 b Ft(a)h(<)g(b)f Fw(b)-5 b(e)45 b(r)-5 b(ational)38 5857 y(numb)g(ers.)44 b(F)-7 b(or)33 b(every)i(c)-5 b(ontinuous)34 b(function)g Ft(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)g Fn(R)45 b Fw(with)34 b Ft(f)11 b Fx(\()p Ft(a)p Fx(\))28 b Fr(\024)g Fx(0)f Fr(\024)i Ft(f)11 b Fx(\()p Ft(b)p Fx(\))p Fw(,)34 b(and)g(every)38 5973 y Ft(k)s Fw(,)h(we)f(c)-5 b(an)34 b(\014nd)h Ft(c)27 b Fr(2)h Fx([)p Ft(a;)17 b(b)p Fx(])36 b Fw(such)f(that)g Fr(j)p Ft(f)11 b Fx(\()p Ft(c)p Fx(\))p Fr(j)26 b(\024)j Fx(2)1899 5937 y Fv(\000)p Fu(k)1996 5973 y Fw(.)p 38 6121 499 4 v 137 6188 a Fm(1)173 6220 y Fg(This)22 b(is)g(pro)n(v)n(ed)f(in)i([5],)g(using)f(the)h(notion)f (of)h(a)f(\\totally)f(b)r(ounded")i(set.)35 b(Ho)n(w)n(ev)n(er,)21 b(the)i(latter)f(is)h(a)f(t)n(yp)r(e-lev)n(el)38 6320 y(2)27 b(concept,)g(whic)n(h)h(w)n(e)f(wish)h(to)f(a)n(v)n(oid.)p eop %%Page: 11 11 11 10 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(In)i(the)g(sequel)h(w)m (e)g(rep)s(eatedly)f(in)m(v)m(ok)m(e)h(Lemma)d(1.13.)73 b(Giv)m(en)43 b Ft(k)s Fx(,)i(let)e Ft(")h Fx(:=)h(2)3389 414 y Fv(\000)p Fu(k)3486 450 y Fx(.)74 b(W)-8 b(e)38 566 y(compare)46 b Ft(f)11 b Fx(\()p Ft(a)p Fx(\))47 b(and)g Ft(f)11 b Fx(\()p Ft(b)p Fx(\))47 b(with)g Fr(\000)p Ft(")52 b(<)g Fr(\000)1730 527 y Fu(")p 1729 543 36 4 v 1729 601 a Fm(2)1821 566 y Fx(and)2037 527 y Fu(")p 2035 543 V 2035 601 a Fm(2)2133 566 y Ft(<)g(")p Fx(,)e(resp)s(ectiv)m (ely)-8 b(.)87 b(If)47 b Fr(\000)p Ft(")52 b(<)g(f)11 b Fx(\()p Ft(a)p Fx(\))47 b(or)38 684 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))28 b Ft(<)f(")p Fx(,)32 b(then)i Fr(j)p Ft(f)11 b Fx(\()p Ft(c)p Fx(\))p Fr(j)26 b Ft(<)i(")k Fx(for)g Ft(c)c Fx(=)f Ft(a)33 b Fx(or)f Ft(c)c Fx(=)f Ft(b)p Fx(;)33 b(whence)i(w)m(e)e(ma)m(y)g(assume)g(that)1271 898 y Ft(f)11 b Fx(\()p Ft(a)p Fx(\))28 b Ft(<)f Fr(\000)1677 831 y Ft(")p 1675 875 49 4 v 1675 966 a Fx(2)1832 898 y(and)2098 831 y Ft(")p 2096 875 V 2096 966 a Fx(2)2183 898 y Ft(<)g(f)11 b Fx(\()p Ft(b)p Fx(\))p Ft(:)38 1128 y Fx(No)m(w)39 b(pic)m(k)g Ft(l)i Fx(so)f(that,)g(for)e(all)f Ft(x;)17 b(y)41 b Fr(2)e Fx([)p Ft(a;)17 b(b)p Fx(],)41 b(if)d Fr(j)p Ft(x)26 b Fr(\000)h Ft(y)t Fr(j)38 b(\024)g Fx(2)2360 1091 y Fv(\000)p Fu(l)2441 1128 y Fx(,)j(then)e Fr(j)p Ft(f)11 b Fx(\()p Ft(x)p Fx(\))26 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(y)t Fx(\))p Fr(j)37 b(\024)i Ft(")p Fx(,)h(and)38 1244 y(divide)e([)p Ft(a;)17 b(b)p Fx(])40 b(in)m(to)e Ft(a)h Fx(=)g Ft(a)1024 1259 y Fm(0)1103 1244 y Ft(<)f(a)1268 1259 y Fm(1)1347 1244 y Ft(<)h Fr(\001)17 b(\001)g(\001)37 b Ft(<)h(a)1782 1259 y Fu(m)1888 1244 y Fx(=)h Ft(b)g Fx(suc)m(h)i(that)e Fr(j)p Ft(a)2607 1259 y Fu(i)p Fv(\000)p Fm(1)2752 1244 y Fr(\000)27 b Ft(a)2907 1259 y Fu(i)2935 1244 y Fr(j)39 b(\024)g Fx(2)3167 1208 y Fv(\000)p Fu(l)3248 1244 y Fx(.)63 b(Compare)38 1360 y(ev)m(ery)37 b Ft(f)11 b Fx(\()p Ft(a)446 1375 y Fu(i)474 1360 y Fx(\))35 b(with)f Fr(\000)859 1321 y Fu(")p 858 1337 36 4 v 858 1394 a Fm(2)936 1360 y Ft(<)1055 1321 y Fu(")p 1054 1337 V 1054 1394 a Fm(2)1099 1360 y Fx(.)51 b(By)36 b(assumption)e Ft(f)11 b Fx(\()p Ft(a)1999 1375 y Fm(0)2038 1360 y Fx(\))32 b Ft(<)g Fr(\000)2304 1321 y Fu(")p 2303 1337 V 2303 1394 a Fm(2)2384 1360 y Fx(and)2587 1321 y Fu(")p 2586 1337 V 2586 1394 a Fm(2)2663 1360 y Ft(<)g(f)11 b Fx(\()p Ft(a)2919 1375 y Fu(m)2985 1360 y Fx(\);)36 b(whence)h(w)m(e)g(can)38 1476 y(\014nd)c Ft(j)38 b Fx(minimal)29 b(suc)m(h)34 b(that)1162 1685 y Ft(f)11 b Fx(\()p Ft(a)1310 1700 y Fu(j)1347 1685 y Fx(\))27 b Ft(<)1527 1617 y(")p 1526 1662 49 4 v 1526 1753 a Fx(2)1682 1685 y(and)120 b Fr(\000)2070 1617 y Ft(")p 2069 1662 V 2069 1753 a Fx(2)2155 1685 y Ft(<)28 b(f)11 b Fx(\()p Ft(a)2407 1700 y Fu(j)t Fm(+1)2533 1685 y Fx(\))p Ft(:)38 1907 y Fx(Finally)-8 b(,)31 b(compare)i Ft(f)11 b Fx(\()p Ft(a)929 1922 y Fu(j)965 1907 y Fx(\))33 b(with)g Fr(\000)p Ft(")c(<)g Fr(\000)1604 1868 y Fu(")p 1603 1884 36 4 v 1603 1941 a Fm(2)1682 1907 y Fx(and)k Ft(f)11 b Fx(\()p Ft(a)2020 1922 y Fu(j)t Fm(+1)2147 1907 y Fx(\))33 b(with)2452 1868 y Fu(")p 2451 1884 V 2451 1941 a Fm(2)2525 1907 y Ft(<)c(")p Fx(.)45 b(If)33 b Fr(\000)p Ft(")c(<)g(f)11 b Fx(\()p Ft(a)3251 1922 y Fu(j)3288 1907 y Fx(\),)33 b(w)m(e)h(ha)m(v)m(e)38 2024 y Fr(j)p Ft(f)11 b Fx(\()p Ft(a)214 2039 y Fu(j)250 2024 y Fx(\))p Fr(j)27 b Ft(<)h(")p Fx(.)42 b(If)28 b Ft(f)11 b Fx(\()p Ft(a)803 2039 y Fu(j)t Fm(+1)930 2024 y Fx(\))27 b Ft(<)h(")p Fx(,)h(w)m(e)h(ha)m(v)m(e)g Fr(j)p Ft(f)11 b Fx(\()p Ft(a)1738 2039 y Fu(j)t Fm(+1)1864 2024 y Fx(\))p Fr(j)27 b Ft(<)h(")p Fx(.)42 b(If)29 b(b)s(oth)f Ft(f)11 b Fx(\()p Ft(a)2644 2039 y Fu(j)2680 2024 y Fx(\))28 b Ft(<)g Fr(\000)2938 1985 y Fu(")p 2937 2001 V 2937 2059 a Fm(2)3011 2024 y Fx(and)3208 1985 y Fu(")p 3207 2001 V 3207 2059 a Fm(2)3280 2024 y Ft(<)f(f)11 b Fx(\()p Ft(a)3531 2039 y Fu(j)t Fm(+1)3658 2024 y Fx(\),)38 2149 y(then)33 b(w)m(e)h(w)m(ould)e(ha)m(v)m(e)i Fr(j)p Ft(f)11 b Fx(\()p Ft(a)1089 2164 y Fu(j)t Fm(+1)1215 2149 y Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)1523 2164 y Fu(j)1559 2149 y Fx(\))p Fr(j)28 b Ft(>)f(")p Fx(,)32 b(con)m(tradicting)g Fr(j)p Ft(a)2534 2164 y Fu(j)t Fm(+1)2683 2149 y Fr(\000)22 b Ft(a)2833 2164 y Fu(j)2870 2149 y Fr(j)27 b(\024)h Fx(2)3079 2113 y Fv(\000)p Fu(l)3160 2149 y Fx(.)p 3682 2090 41 4 v 3682 2146 4 56 v 3720 2146 V 3682 2149 41 4 v 38 2352 a Fw(A)n(lternative)34 b(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)32 b(giv)m(e)g(a)f(di\013eren)m(t)h(pro)s(of,)f(whic)m(h) h(more)f(directly)f(mak)m(es)j(use)f(of)f(the)h(fact)38 2468 y(that)g(our)h(con)m(tin)m(uous)g(functions)f(come)h(with)f (witnessing)g(data.)137 2585 y(W)-8 b(e)44 b(ma)m(y)e(assume)i Ft(f)11 b Fx(\()p Ft(a)p Fx(\))45 b Ft(<)g Fr(\000)p Fx(2)1363 2548 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1593 2585 y Fx(and)e(2)1842 2548 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2075 2585 y Ft(<)i(f)11 b Fx(\()p Ft(b)p Fx(\))43 b(\(see)h(ab)s(o)m(v)m(e\).)75 b(Divide)41 b([)p Ft(a;)17 b(b)p Fx(])44 b(in)m(to)38 2704 y Ft(a)28 b Fx(=)f Ft(a)271 2719 y Fm(0)339 2704 y Ft(<)g(a)493 2719 y Fm(1)560 2704 y Ft(<)h Fr(\001)17 b(\001)g(\001)26 b Ft(<)h(a)962 2719 y Fu(m)1057 2704 y Fx(=)g Ft(b)i Fx(suc)m(h)g(that)e Fr(j)p Ft(a)1730 2719 y Fu(i)p Fv(\000)p Fm(1)1861 2704 y Fr(\000)12 b Ft(a)2001 2719 y Fu(i)2030 2704 y Fr(j)27 b(\024)h Fx(2)2239 2668 y Fv(\000)p Fu(!)2338 2680 y Fk(f)2377 2668 y Fm(\()p Fu(k)r Fm(+1\))2565 2704 y Fx(.)42 b(Consider)28 b(all)e(\014nitely)h(man)m(y)1309 2891 y Ft(h)p Fx(\()p Ft(a)1454 2906 y Fu(i)1482 2891 y Ft(;)17 b(n)1584 2906 y Fm(0)1623 2891 y Fx(\))98 b(for)32 b Ft(i)c Fx(=)f(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(m)p Fx(,)38 3078 y(with)37 b Ft(n)323 3093 y Fm(0)398 3078 y Fx(:=)f Ft(\013)599 3093 y Fu(f)644 3078 y Fx(\()p Ft(k)28 b Fx(+)e(1\).)57 b(Pic)m(k)38 b Ft(j)43 b Fx(suc)m(h)c(that)e Ft(h)p Fx(\()p Ft(a)1926 3093 y Fu(j)t Fv(\000)p Fm(1)2053 3078 y Ft(;)17 b(n)2155 3093 y Fm(0)2194 3078 y Fx(\))36 b Fr(\024)g Fx(0)f Fr(\024)i Ft(h)p Fx(\()p Ft(a)2724 3093 y Fu(j)2760 3078 y Ft(;)17 b(n)2862 3093 y Fm(0)2902 3078 y Fx(\);)39 b(this)e(can)h(b)s(e)f(done)38 3196 y(b)s(ecause)i Ft(f)11 b Fx(\()p Ft(a)p Fx(\))36 b Ft(<)g Fr(\000)p Fx(2)864 3160 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1090 3196 y Fx(and)h(2)1333 3160 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1557 3196 y Ft(<)f(f)11 b Fx(\()p Ft(b)p Fx(\).)59 b(W)-8 b(e)38 b(sho)m(w)h Fr(j)p Ft(f)11 b Fx(\()p Ft(a)2527 3211 y Fu(j)2563 3196 y Fx(\))p Fr(j)36 b(\024)g Fx(2)2827 3160 y Fv(\000)p Fu(k)2925 3196 y Fx(;)k(for)d(this)g(it)g(clearly)38 3314 y(su\016ces)d(to)f(sho)m(w)g Fr(j)p Ft(h)p Fx(\()p Ft(a)903 3329 y Fu(j)940 3314 y Ft(;)17 b(n)p Fx(\))p Fr(j)27 b(\024)h Fx(2)1289 3278 y Fv(\000)p Fu(k)1419 3314 y Fx(for)k Ft(n)c Fr(\025)g Ft(n)1817 3329 y Fm(0)1857 3314 y Fx(.)43 b(No)m(w)540 3509 y Fr(j)p Ft(h)p Fx(\()p Ft(a)713 3524 y Fu(j)749 3509 y Ft(;)17 b(n)p Fx(\))p Fr(j)27 b(\024)1049 3424 y Fj(\014)1049 3484 y(\014)1083 3509 y Ft(h)p Fx(\()p Ft(a)1228 3524 y Fu(j)1264 3509 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)p Fx(\()p Ft(a)1671 3524 y Fu(j)1708 3509 y Ft(;)17 b(n)1810 3524 y Fm(0)1849 3509 y Fx(\))1887 3424 y Fj(\014)1887 3484 y(\014)1942 3509 y Fx(+)2041 3424 y Fj(\014)2041 3484 y(\014)2074 3509 y Ft(h)p Fx(\()p Ft(a)2219 3524 y Fu(j)2256 3509 y Ft(;)g(n)2358 3524 y Fm(0)2397 3509 y Fx(\))2435 3424 y Fj(\014)2435 3484 y(\014)2496 3509 y Fr(\024)28 b Fx(2)2650 3467 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2860 3509 y Fx(+)22 b(2)3007 3467 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)3194 3509 y Ft(;)38 3696 y Fx(where)33 b(the)f(\014rst)h(estimate)e(holds)g(b)m(y)i(the)f (c)m(hoice)g(of)g Ft(n)2091 3711 y Fm(0)2131 3696 y Fx(,)g(and)g(the)g (second)h(one)f(follo)m(ws)f(from)f(the)38 3812 y(c)m(hoice)j(of)f Ft(a)490 3827 y Fu(j)559 3812 y Fx(and)h Fr(j)p Ft(h)p Fx(\()p Ft(a)922 3827 y Fu(i)p Fv(\000)p Fm(1)1040 3812 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)p Fx(\()p Ft(a)1447 3827 y Fu(i)1475 3812 y Ft(;)17 b(n)p Fx(\))p Fr(j)27 b(\024)i Fx(2)1825 3776 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2012 3812 y Fx(.)p 3682 3753 V 3682 3808 4 56 v 3720 3808 V 3682 3811 41 4 v 137 4015 a(A)d(function)f Ft(f)d Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)g Fn(R)36 b Fx(is)25 b Fw(lo)-5 b(c)g(al)5 b(ly)28 b(nonc)-5 b(onstant)34 b Fx(whenev)m(er)28 b(if)d Ft(a)j Fr(\024)g Ft(a)2852 3978 y Fv(0)2903 4015 y Ft(<)f(b)3047 3978 y Fv(0)3099 4015 y Fr(\024)h Ft(b)e Fx(and)g Ft(z)k Fx(is)25 b(an)38 4131 y(arbitrary)h(real,)h(then)h Ft(f)11 b Fx(\()p Ft(x)p Fx(\))28 b Fr(6)p Fx(=)f Ft(z)32 b Fx(for)27 b(some)g(real)f Ft(x)i Fr(2)g Fx([)p Ft(a)2096 4095 y Fv(0)2120 4131 y Ft(;)17 b(b)2205 4095 y Fv(0)2228 4131 y Fx(].)42 b(Note)27 b(that)g(if)f Ft(f)38 b Fx(is)26 b(con)m(tin)m(uous,)j(then)38 4247 y(there)39 b(also)e(is)g(a)h(rational)d(with)j(that)g(prop)s(ert)m (y)-8 b(.)60 b(Strictly)37 b(monotonous)h(functions)g(are)g(clearly)38 4363 y(lo)s(cally)30 b(nonconstan)m(t,)j(and)g(so)g(are)f(nonconstan)m (t)i(real)e(p)s(olynomials.)38 4515 y Fp(Prop)s(osition)41 b(3.13)h Fx(\(In)m(termediate)37 b(V)-8 b(alue)37 b(Theorem\))p Fp(.)44 b Fw(L)-5 b(et)39 b Ft(a)d(<)f(b)40 b Fw(b)-5 b(e)38 b(r)-5 b(ational)39 b(numb)-5 b(ers.)56 b(If)38 4631 y Ft(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)f Fn(R)45 b Fw(is)34 b(c)-5 b(ontinuous)33 b(with)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))27 b Ft(<)h Fx(0)f Ft(<)h(f)11 b Fx(\()p Ft(b)p Fx(\))p Fw(,)33 b(and)h(lo)-5 b(c)g(al)5 b(ly)33 b(nonc)-5 b(onstant,)33 b(then)g(we)g(c)-5 b(an)38 4747 y(\014nd)34 b Ft(x)28 b Fr(2)g Fx([)p Ft(a;)17 b(b)p Fx(])36 b Fw(with)e Ft(f)11 b Fx(\()p Ft(x)p Fx(\))28 b(=)g(0)p Fw(.)38 4950 y(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)33 b(construct)h(t)m(w)m(o)f(sequences)j(\()p Ft(c)1640 4965 y Fu(n)1687 4950 y Fx(\))1725 4965 y Fu(n)1804 4950 y Fx(and)d(\()p Ft(d)2083 4965 y Fu(n)2129 4950 y Fx(\))2167 4965 y Fu(n)2247 4950 y Fx(of)f(rationals)e(suc)m(h)k(that)f(for)f(all) e Ft(m)824 5137 y(a)e Fx(=)g Ft(c)1049 5152 y Fm(0)1116 5137 y Fr(\024)g Ft(c)1263 5152 y Fm(1)1330 5137 y Fr(\024)g(\001)17 b(\001)g(\001)26 b(\024)i Ft(c)1726 5152 y Fu(m)1821 5137 y Ft(<)f(d)1975 5152 y Fu(m)2069 5137 y Fr(\024)h(\001)17 b(\001)g(\001)26 b(\024)i Ft(d)2474 5152 y Fm(1)2541 5137 y Fr(\024)g Ft(d)2697 5152 y Fm(0)2764 5137 y Fx(=)g Ft(b;)-2898 b Fx(\(7\))824 5288 y Ft(f)11 b Fx(\()p Ft(c)963 5303 y Fu(m)1030 5288 y Fx(\))27 b Ft(<)h Fx(0)f Ft(<)h(f)11 b Fx(\()p Ft(d)1527 5303 y Fu(m)1593 5288 y Fx(\))p Ft(;)-1620 b Fx(\(8\))824 5496 y Ft(d)875 5511 y Fu(m)964 5496 y Fr(\000)22 b Ft(c)1105 5511 y Fu(m)1200 5496 y Fr(\024)1305 5386 y Fj(\020)1374 5429 y Fx(2)p 1374 5473 49 4 v 1374 5564 a(3)1433 5386 y Fj(\021)1493 5408 y Fu(m)1559 5496 y Fx(\()p Ft(b)h Fr(\000)f Ft(a)p Fx(\))p Ft(:)-1838 b Fx(\(9\))38 5718 y(Let)39 b Ft(c)261 5733 y Fm(0)300 5718 y Ft(;)17 b(:)g(:)g(:)f(;)h(c)561 5733 y Fu(m)666 5718 y Fx(and)39 b Ft(d)913 5733 y Fm(0)952 5718 y Ft(;)17 b(:)g(:)g(:)f(;)h(d)1222 5733 y Fu(m)1326 5718 y Fx(b)s(e)39 b(already)g(constructed)h(suc)m(h)g(that)f(\(7\)-\(9\))e(hold.)62 b(Let)39 b Ft(c)f Fx(=)38 5834 y Ft(c)80 5849 y Fu(m)148 5834 y Fx(+)235 5795 y Fm(1)p 235 5811 36 4 v 235 5869 a(3)280 5834 y Fx(\()p Ft(d)369 5849 y Fu(m)437 5834 y Fr(\000)q Ft(c)557 5849 y Fu(m)625 5834 y Fx(\))22 b(and)h Ft(d)k Fx(=)g Ft(c)1088 5849 y Fu(m)1156 5834 y Fx(+)1243 5795 y Fm(2)p 1244 5811 V 1244 5869 a(3)1289 5834 y Fx(\()p Ft(d)1378 5849 y Fu(m)1446 5834 y Fr(\000)q Ft(c)1566 5849 y Fu(m)1633 5834 y Fx(\).)40 b(By)23 b(assumption)f(w)m (e)i(ha)m(v)m(e)f(a)g(rational)d Ft(a)3211 5849 y Fu(m)3305 5834 y Fr(2)28 b Fx([)p Ft(c)3468 5849 y Fu(m)3535 5834 y Ft(;)17 b(d)3630 5849 y Fu(m)3696 5834 y Fx(])38 5952 y(with)24 b Ft(f)11 b Fx(\()p Ft(a)400 5967 y Fu(m)466 5952 y Fx(\))28 b Fr(6)p Fx(=)f(0.)41 b(In)24 b(case)h(0)j Ft(<)f(f)11 b Fx(\()p Ft(a)1391 5967 y Fu(m)1458 5952 y Fx(\))24 b(let)f Ft(c)1694 5967 y Fu(m)p Fm(+1)1879 5952 y Fx(=)k Ft(c)2024 5967 y Fu(m)2115 5952 y Fx(and)d Ft(d)2347 5967 y Fu(m)p Fm(+1)2531 5952 y Fx(=)k Ft(a)2686 5967 y Fu(m)2753 5952 y Fx(,)e(and)e(in)f(case)i Ft(f)11 b Fx(\()p Ft(a)3438 5967 y Fu(m)3505 5952 y Fx(\))28 b Ft(<)f Fx(0)38 6068 y(let)36 b Ft(c)225 6083 y Fu(m)p Fm(+1)417 6068 y Fx(=)e Ft(a)578 6083 y Fu(m)682 6068 y Fx(and)j Ft(y)924 6083 y Fu(m)p Fm(+1)1115 6068 y Fx(=)e Ft(y)1274 6083 y Fu(m)1340 6068 y Fx(.)56 b(Let)37 b Ft(x;)17 b(y)40 b Fx(b)s(e)d(giv)m(en)g(b)m(y)h(the)f(mo)s(dulus)e Fs(Ca)n(uchy)40 b Fx(sequences)38 6184 y(\()p Ft(c)118 6199 y Fu(n)165 6184 y Fx(\))203 6199 y Fu(n)284 6184 y Fx(and)35 b(\()p Ft(d)565 6199 y Fu(n)612 6184 y Fx(\))650 6199 y Fu(n)732 6184 y Fx(of)f(rationals.)48 b(As)36 b Ft(f)45 b Fx(is)34 b(con)m(tin)m(uous,)j Ft(f)11 b Fx(\()p Ft(x)p Fx(\))31 b(=)h(0)f(=)g Ft(f)11 b Fx(\()p Ft(y)t Fx(\))34 b(for)g(the)h(real)f(n)m(um)m(b)s(er)38 6300 y Ft(x)28 b Fx(=)f Ft(y)t Fx(.)p 3682 6241 41 4 v 3682 6297 4 56 v 3720 6297 V 3682 6300 41 4 v eop %%Page: 12 12 12 11 bop 1414 450 a Fx(4.)48 b Fs(Differentia)-7 b(tion)38 624 y Fp(De\014nition)25 b(4.1.)35 b Fx(Let)24 b Ft(f)5 b(;)17 b(g)e Fx(:)33 b Ft(I)i Fr(!)27 b Fn(R)35 b Fx(b)s(e)23 b(con)m(tin)m(uous.)42 b Ft(g)26 b Fx(is)d(called)g Fw(derivative)30 b Fx(of)23 b Ft(f)34 b Fx(with)23 b(mo)s(dulus)38 741 y Ft(\016)81 756 y Fu(f)137 741 y Fx(:)34 b Fn(N)42 b Fr(!)28 b Fn(N)47 b Fx(of)32 b(di\013eren)m(tiabilit)m(y)e(if)i(for)g Ft(a;)17 b(b)28 b Fr(2)g Ft(I)40 b Fx(with)32 b Ft(a)c(<)g(b)p Fx(,)626 941 y Ft(b)g Fr(\024)g Ft(a)22 b Fx(+)h(2)1021 900 y Fv(\000)p Fu(\016)1107 912 y Fk(f)1146 900 y Fm(\()p Fu(k)r Fm(\))1271 941 y Fr(!)1399 856 y Fj(\014)1399 916 y(\014)1432 941 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)g Ft(g)t Fx(\()p Ft(a)p Fx(\)\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))2505 856 y Fj(\014)2505 916 y(\014)2566 941 y Fr(\024)28 b Fx(2)2720 900 y Fv(\000)p Fu(k)2817 941 y Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))p Ft(:)38 1130 y(f)43 b Fx(is)32 b(said)g(to)g(b)s(e)h Fw(di\013er)-5 b(entiable)39 b Fx(on)32 b Ft(I)41 b Fx(and)32 b Ft(g)k Fx(is)c(called)g(a)g Fw(derivative)39 b Fx(of)32 b Ft(f)43 b Fx(on)33 b Ft(I)8 b Fx(.)137 1283 y(T)-8 b(o)33 b(sa)m(y)h(that)e Ft(g)k Fx(is)c(a)g(deriv)-5 b(ativ)m(e)32 b(of)g Ft(f)44 b Fx(w)m(e)33 b(write)1064 1526 y Ft(g)e Fx(=)c Ft(f)1304 1485 y Fv(0)1327 1526 y Ft(;)115 b(g)31 b Fx(=)c Ft(D)s(f)5 b(;)114 b Fx(or)98 b Ft(g)t Fx(\()p Ft(x)p Fx(\))27 b(=)2435 1459 y Ft(d)-16 b(f)11 b Fx(\()p Ft(x)p Fx(\))p 2435 1503 225 4 v 2495 1595 a Ft(dx)2670 1526 y(:)38 1745 y Fx(If)32 b Ft(f)44 b Fx(has)33 b(t)m(w)m(o)g(deriv)-5 b(ativ)m(es,)32 b(then)i(clearly)d(they)j(are)e(equal)h(functions.)137 1861 y(F)-8 b(or)33 b(example,)f(a)h(constan)m(t)h(function)e(has)i (deriv)-5 b(ativ)m(e)32 b(0,)h(and)g(the)h(iden)m(tit)m(y)e(function)h (has)g(the)38 1978 y(constan)m(t)g(1)g(function)f(as)g(deriv)-5 b(ativ)m(e.)137 2094 y(W)d(e)33 b(sho)m(w)h(that)e(a)h(b)s(ound)g(on)f (the)h(deriv)-5 b(ativ)m(e)32 b(of)g Ft(f)43 b Fx(serv)m(es)35 b(as)e(a)f Fs(Lipschitz)g Fx(constan)m(t)i(of)e Ft(f)11 b Fx(:)38 2247 y Fp(Lemma)28 b(4.2.)36 b Fw(L)-5 b(et)29 b Ft(f)22 b Fx(:)33 b Ft(I)i Fr(!)28 b Fn(R)39 b Fw(b)-5 b(e)27 b(c)-5 b(ontinuous)28 b(with)g(derivative)f Ft(f)2534 2211 y Fv(0)2557 2247 y Fw(.)43 b(Assume)27 b(that)i Ft(f)3238 2211 y Fv(0)3289 2247 y Fw(is)f(b)-5 b(ounde)g(d)38 2363 y(on)34 b Ft(I)43 b Fw(by)35 b Ft(M)10 b Fw(.)46 b(Then)34 b(for)g Ft(a;)17 b(b)28 b Fr(2)g Ft(I)43 b Fw(with)35 b Ft(a)28 b(<)f(b)p Fw(,)1328 2468 y Fj(\014)1328 2527 y(\014)1361 2552 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1845 2468 y Fj(\014)1845 2527 y(\014)1905 2552 y Fr(\024)29 b Ft(M)10 b Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))p Ft(:)38 2758 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Giv)m(en)32 b Ft(k)f Fr(2)d Fn(N)9 b Fx(,)39 b(it)31 b(su\016ces)k(to)d(pro)m(v)m(e)1208 2863 y Fj(\014)1208 2923 y(\014)1241 2948 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1725 2863 y Fj(\014)1725 2923 y(\014)1786 2948 y Fr(\024)28 b Ft(M)10 b Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))f(+)g(2)2455 2906 y Fv(\000)p Fu(k)38 3146 y Fx(Cho)s(ose)39 b Ft(l)i Fx(suc)m(h)f(that)e(2)945 3110 y Fv(\000)p Fu(l)1026 3146 y Fx(\()p Ft(b)26 b Fr(\000)h Ft(a)p Fx(\))38 b Fr(\024)g Fx(2)1526 3110 y Fv(\000)p Fu(k)1624 3146 y Fx(,)i(and)e(let)g Ft(a)g Fx(=)g Ft(a)2287 3161 y Fm(0)2365 3146 y Ft(<)f(a)2529 3161 y Fm(1)2607 3146 y Ft(<)h Fr(\001)17 b(\001)g(\001)36 b Ft(<)h(a)3039 3161 y Fu(n)3124 3146 y Fx(=)h Ft(b)h Fx(suc)m(h)h(that)38 3265 y Ft(a)89 3280 y Fu(i)p Fm(+1)235 3265 y Fr(\024)28 b Ft(a)391 3280 y Fu(i)442 3265 y Fx(+)22 b(2)589 3229 y Fv(\000)p Fu(\016)675 3241 y Fk(f)714 3229 y Fm(\()p Fu(l)q Fm(\))795 3265 y Fx(,)33 b(where)h Ft(\016)1180 3280 y Fu(f)1258 3265 y Fx(is)e(the)h(mo)s(dulus)e(of)h(di\013eren)m (tiabilit)m(y)e(of)i Ft(f)11 b Fx(.)44 b(Then)496 3375 y Fj(\014)496 3434 y(\014)529 3459 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))23 b Fr(\000)f Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1013 3375 y Fj(\014)1013 3434 y(\014)524 3713 y Fx(=)627 3598 y Fj(\014)627 3658 y(\014)627 3718 y(\014)666 3589 y Fu(n)p Fv(\000)p Fm(1)661 3618 y Fj(X)676 3828 y Fu(i)p Fm(=0)821 3713 y Ft(f)g Fx(\()p Ft(a)969 3728 y Fu(i)p Fm(+1)1087 3713 y Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)1395 3728 y Fu(i)1423 3713 y Fx(\))1461 3598 y Fj(\014)1461 3658 y(\014)1461 3718 y(\014)524 4053 y Fx(=)627 3938 y Fj(\014)627 3998 y(\014)627 4058 y(\014)666 3928 y Fu(n)p Fv(\000)p Fm(1)661 3958 y Fj(X)676 4168 y Fu(i)p Fm(=0)821 4053 y Ft(f)880 4012 y Fv(0)903 4053 y Fx(\()p Ft(a)992 4068 y Fu(i)1021 4053 y Fx(\)\()p Ft(a)1148 4068 y Fu(i)p Fm(+1)1288 4053 y Fr(\000)23 b Ft(a)1439 4068 y Fu(i)1467 4053 y Fx(\))f(+)1631 3928 y Fu(n)p Fv(\000)p Fm(1)1625 3958 y Fj(X)1640 4168 y Fu(i)p Fm(=0)1786 4053 y Ft(f)11 b Fx(\()p Ft(a)1934 4068 y Fu(i)p Fm(+1)2052 4053 y Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)2360 4068 y Fu(i)2388 4053 y Fx(\))22 b Fr(\000)g Ft(f)2606 4012 y Fv(0)2629 4053 y Fx(\()p Ft(a)2718 4068 y Fu(i)2747 4053 y Fx(\)\()p Ft(a)2874 4068 y Fu(i)p Fm(+1)3014 4053 y Fr(\000)h Ft(a)3165 4068 y Fu(i)3193 4053 y Fx(\))3231 3938 y Fj(\014)3231 3998 y(\014)3231 4058 y(\014)524 4303 y Fr(\024)28 b Ft(M)10 b Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))f(+)g(2)1193 4262 y Fv(\000)p Fu(l)1274 4303 y Fx(\()p Ft(b)h Fr(\000)f Ft(a)p Fx(\))p Ft(:)38 4493 y Fx(This)33 b(pro)m(v)m(es)h(the)f(claim.) p 3682 4433 41 4 v 3682 4489 4 56 v 3720 4489 V 3682 4492 41 4 v 38 4698 a Fp(Lemma)50 b(4.3.)d Fw(L)-5 b(et)46 b Ft(f)5 b(;)17 b(g)e Fx(:)36 b Ft(I)55 b Fr(!)46 b Fn(R)56 b Fw(b)-5 b(e)45 b(c)-5 b(ontinuous)45 b(with)g(derivatives)f Ft(f)2840 4662 y Fv(0)2863 4698 y Ft(;)17 b(g)2958 4662 y Fv(0)3026 4698 y Fw(of)45 b(mo)-5 b(duli)44 b Ft(\016)3520 4713 y Fu(f)3566 4698 y Ft(;)17 b(\016)3653 4713 y Fu(g)3693 4698 y Fw(.)38 4815 y(Then)1499 4963 y Fx(\()p Ft(f)33 b Fx(+)22 b Ft(g)t Fx(\))1805 4922 y Fv(0)1855 4963 y Fx(:=)28 b Ft(f)2045 4922 y Fv(0)2090 4963 y Fx(+)22 b Ft(g)2239 4922 y Fv(0)38 5132 y Fw(is)34 b(a)h(derivative)f(of)h Ft(f)e Fx(+)22 b Ft(g)38 b Fw(with)c(mo)-5 b(dulus)1090 5322 y Ft(\016)1133 5337 y Fu(f)7 b Fm(+)p Fu(g)1269 5322 y Fx(\()p Ft(k)s Fx(\))28 b(:=)g(max)1739 5241 y Fj(\000)1785 5322 y Ft(\016)1828 5337 y Fu(f)1873 5322 y Fx(\()p Ft(k)d Fx(+)d(1\))p Ft(;)17 b(\016)2259 5337 y Fu(g)2299 5322 y Fx(\()p Ft(k)25 b Fx(+)d(1\))2598 5241 y Fj(\001)2644 5322 y Ft(:)38 5531 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(a)28 b(<)f(b)h Fr(\024)g Ft(a)23 b Fx(+)f(2)1088 5495 y Fv(\000)p Fu(m)1209 5531 y Fx(.)43 b(Then)595 5632 y Fj(\014)595 5692 y(\014)628 5716 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))23 b(+)f Ft(g)t Fx(\()p Ft(b)p Fx(\))f Fr(\000)i Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(g)t Fx(\()p Ft(a)p Fx(\))e Fr(\000)i Fx(\()p Ft(f)1918 5675 y Fv(0)1941 5716 y Fx(\()p Ft(a)p Fx(\))f Fr(\000)h Ft(g)2241 5675 y Fv(0)2263 5716 y Fx(\()p Ft(a)p Fx(\)\)\()p Ft(b)g Fr(\000)f Ft(a)p Fx(\))2718 5632 y Fj(\014)2718 5692 y(\014)623 5907 y Fr(\024)728 5823 y Fj(\014)728 5882 y(\014)761 5907 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(f)1426 5866 y Fv(0)1449 5907 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))1866 5823 y Fj(\014)1866 5882 y(\014)1921 5907 y Fx(+)2019 5823 y Fj(\014)2019 5882 y(\014)2053 5907 y Ft(g)t Fx(\()p Ft(b)p Fx(\))e Fr(\000)i Ft(g)t Fx(\()p Ft(a)p Fx(\))f Fr(\000)g Ft(g)2692 5866 y Fv(0)2715 5907 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))3132 5793 y Fj(\014)3132 5853 y(\014)3132 5912 y(\014)623 6097 y Fr(\024)28 b Fx(2)777 6056 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)965 6097 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))f(+)g(2)1424 6056 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1612 6097 y Fx(\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))p Ft(:)38 6292 y Fx(for)32 b Ft(m)c Fr(\025)g Ft(\016)448 6307 y Fu(f)494 6292 y Fx(\()p Ft(k)d Fx(+)d(1\))p Ft(;)17 b(\016)880 6307 y Fu(g)919 6292 y Fx(\()p Ft(k)26 b Fx(+)c(1\).)p 3682 6233 V 3682 6289 4 56 v 3720 6289 V 3682 6292 41 4 v eop %%Page: 13 13 13 12 bop 38 450 a Fp(Lemma)50 b(4.4.)d Fw(L)-5 b(et)46 b Ft(f)5 b(;)17 b(g)e Fx(:)36 b Ft(I)55 b Fr(!)46 b Fn(R)56 b Fw(b)-5 b(e)45 b(c)-5 b(ontinuous)45 b(with)g(derivatives)f Ft(f)2840 414 y Fv(0)2863 450 y Ft(;)17 b(g)2958 414 y Fv(0)3026 450 y Fw(of)45 b(mo)-5 b(duli)44 b Ft(\016)3520 465 y Fu(f)3566 450 y Ft(;)17 b(\016)3653 465 y Fu(g)3693 450 y Fw(.)38 566 y(Then)1505 715 y Fx(\()p Ft(f)11 b(g)t Fx(\))1691 674 y Fv(0)1741 715 y Fx(:=)27 b Ft(f)1930 674 y Fv(0)1953 715 y Ft(g)f Fx(+)c Ft(f)11 b(g)2234 674 y Fv(0)38 885 y Fw(is)34 b(a)h(derivative)f(of)h Ft(f)11 b(g)38 b Fw(with)c(mo)-5 b(dulus)644 1074 y Ft(\016)687 1089 y Fu(f)7 b(g)769 1074 y Fx(\()p Ft(k)s Fx(\))27 b(:=)h(max)1238 993 y Fj(\000)1284 1074 y Ft(!)1345 1089 y Fu(g)1385 1074 y Fx(\()p Ft(r)d Fx(+)d Ft(k)j Fx(+)d(1\))p Ft(;)17 b(\016)1938 1089 y Fu(f)1983 1074 y Fx(\()p Ft(k)25 b Fx(+)d Ft(q)k Fx(+)c(2\))p Ft(;)17 b(\016)2536 1089 y Fu(g)2576 1074 y Fx(\()p Ft(k)25 b Fx(+)d Ft(p)g Fx(+)g(2\))3044 993 y Fj(\001)3089 1074 y Ft(;)38 1264 y Fw(wher)-5 b(e)34 b Fx(2)362 1228 y Fu(r)400 1264 y Ft(;)17 b Fx(2)493 1228 y Fu(p)532 1264 y Ft(;)g Fx(2)625 1228 y Fu(q)697 1264 y Fw(ar)-5 b(e)35 b(upp)-5 b(er)35 b(b)-5 b(ounds)34 b(for)h Ft(f)1667 1228 y Fv(0)1690 1264 y Ft(;)17 b(f)5 b(;)17 b(g)37 b Fw(in)e Ft(I)8 b Fw(,)35 b(r)-5 b(esp)g(e)g(ctively.)38 1470 y(Pr)g(o)g(of.)41 b Fx(Let)33 b Ft(a)28 b(<)f(b)h Fr(\024)g Ft(a)23 b Fx(+)f(2)1088 1434 y Fv(\000)p Fu(m)1209 1470 y Fx(.)43 b(Then,)34 b(using)f(Lemma)e(4.2)425 1571 y Fj(\014)425 1631 y(\014)458 1656 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))p Ft(g)t Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(a)p Fx(\))p Ft(g)t Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)g Ft(f)1467 1614 y Fv(0)1490 1656 y Fx(\()p Ft(a)p Fx(\))p Ft(g)t Fx(\()p Ft(a)p Fx(\)\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))f Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))p Ft(g)2444 1614 y Fv(0)2466 1656 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\))2883 1571 y Fj(\014)2883 1631 y(\014)452 1817 y Fx(=)556 1732 y Fj(\014)556 1792 y(\014)589 1736 y(\000)635 1817 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1119 1736 y Fj(\001)1164 1817 y Ft(g)t Fx(\()p Ft(b)p Fx(\))22 b(+)g Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1638 1736 y Fj(\000)1683 1817 y Ft(g)t Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(g)t Fx(\()p Ft(a)p Fx(\))2151 1736 y Fj(\001)2218 1817 y Fr(\000)620 1978 y Ft(f)679 1937 y Fv(0)702 1978 y Fx(\()p Ft(a)p Fx(\))p Ft(g)t Fx(\()p Ft(a)p Fx(\)\()p Ft(b)f Fr(\000)g Ft(a)p Fx(\))h Fr(\000)f Ft(f)11 b Fx(\()p Ft(a)p Fx(\))p Ft(g)1655 1937 y Fv(0)1678 1978 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))2095 1893 y Fj(\014)2095 1953 y(\014)452 2139 y Fx(=)556 2054 y Fj(\014)556 2114 y(\014)589 2058 y(\000)635 2139 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1119 2058 y Fj(\001)o(\000)1210 2139 y Ft(g)t Fx(\()p Ft(b)p Fx(\))21 b Fr(\000)i Ft(g)t Fx(\()p Ft(a)p Fx(\))1677 2058 y Fj(\001)1744 2139 y Fx(+)1842 2058 y Fj(\000)1888 2139 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)g Ft(f)2552 2098 y Fv(0)2575 2139 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)h Fr(\000)g Ft(a)p Fx(\))2993 2058 y Fj(\001)3038 2139 y Ft(g)t Fx(\()p Ft(a)p Fx(\))f(+)620 2300 y Ft(f)11 b Fx(\()p Ft(a)p Fx(\))806 2219 y Fj(\000)851 2300 y Ft(g)t Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(g)t Fx(\()p Ft(a)p Fx(\))e Fr(\000)i Ft(g)1491 2259 y Fv(0)1513 2300 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)g Fr(\000)g Ft(a)p Fx(\))1931 2219 y Fj(\001)1976 2215 y(\014)1976 2275 y(\014)452 2464 y Fr(\024)29 b Fx(2)607 2422 y Fu(r)644 2464 y Fx(\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\)2)983 2422 y Fv(\000)p Fu(r)r Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1282 2464 y Fx(+)g(2)1429 2422 y Fv(\000)p Fu(k)r Fv(\000)p Fu(q)r Fv(\000)p Fm(2)1705 2464 y Fx(\()p Ft(b)h Fr(\000)f Ft(a)p Fx(\))1995 2379 y Fj(\014)1995 2439 y(\014)2029 2464 y Ft(g)t Fx(\()p Ft(a)p Fx(\))2207 2379 y Fj(\014)2207 2439 y(\014)2261 2464 y Fx(+)2359 2379 y Fj(\014)2359 2439 y(\014)2393 2464 y Ft(f)11 b Fx(\()p Ft(a)p Fx(\))2579 2379 y Fj(\014)2579 2439 y(\014)2612 2464 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\)2)2951 2422 y Fv(\000)p Fu(k)r Fv(\000)p Fu(p)p Fv(\000)p Fm(2)452 2627 y Fr(\024)29 b Fx(2)607 2586 y Fv(\000)p Fu(k)704 2627 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))38 2823 y(for)32 b Ft(m)c Fr(\025)g Ft(!)466 2838 y Fu(g)506 2823 y Fx(\()p Ft(r)d Fx(+)d Ft(k)j Fx(+)d(1\))p Ft(;)17 b(\016)1059 2838 y Fu(f)1104 2823 y Fx(\()p Ft(k)25 b Fx(+)d Ft(q)k Fx(+)c(2\))p Ft(;)17 b(\016)1657 2838 y Fu(g)1697 2823 y Fx(\()p Ft(k)25 b Fx(+)d Ft(p)g Fx(+)g(2\).)p 3682 2764 41 4 v 3682 2819 4 56 v 3720 2819 V 3682 2822 41 4 v 38 3029 a Fp(Lemma)27 b(4.5.)36 b Fw(L)-5 b(et)27 b Ft(g)15 b Fx(:)33 b Ft(I)i Fr(!)28 b Fn(R)38 b Fw(b)-5 b(e)27 b(c)-5 b(ontinuous)26 b(with)h(derivative)g Ft(g)2511 2993 y Fv(0)2561 3029 y Fw(of)f(mo)-5 b(dulus)27 b Ft(\016)3081 3044 y Fu(g)3122 3029 y Fw(,)h(and)f Fr(j)p Ft(g)3441 2993 y Fv(0)3463 3029 y Fx(\()p Ft(a)p Fx(\))p Fr(j)g(\024)38 3145 y Fx(2)87 3109 y Fu(q)159 3145 y Fw(for)35 b(al)5 b(l)35 b Ft(a)28 b Fr(2)g Ft(I)8 b Fw(.)44 b(Mor)-5 b(e)g(over)35 b(assume)f(that)h Fx(2)1771 3109 y Fv(\000)p Fu(p)1893 3145 y Fr(\024)28 b(j)p Ft(g)t Fx(\()p Ft(a)p Fx(\))p Fr(j)34 b Fw(for)h(al)5 b(l)34 b Ft(a)28 b Fr(2)g Ft(I)8 b Fw(.)45 b(Then)1538 3283 y Fj(\020)1608 3326 y Fx(1)p 1607 3371 51 4 v 1607 3462 a Ft(g)1668 3283 y Fj(\021)1727 3305 y Fv(0)1778 3394 y Fx(:=)28 b Fr(\000)2004 3326 y Ft(g)2055 3290 y Fv(0)2078 3326 y Fx(\()p Ft(a)p Fx(\))p 1996 3371 217 4 v 1996 3462 a Ft(g)t Fx(\()p Ft(a)p Fx(\))2174 3433 y Fm(2)38 3651 y Fw(is)34 b(a)h(derivative)f(of)799 3611 y Fm(1)p 799 3628 36 4 v 799 3685 a Fu(g)879 3651 y Fw(with)h(mo)-5 b(dulus)839 3864 y Ft(\016)893 3862 y Fh(1)p 892 3874 33 3 v 892 3915 a Fk(g)938 3864 y Fx(\()p Ft(k)s Fx(\))28 b(:=)f(max)1408 3784 y Fj(\000)1454 3864 y Ft(\016)1497 3879 y Fu(g)1537 3864 y Fx(\()p Ft(k)e Fx(+)d(2)p Ft(p)g Fx(+)g(1\))p Ft(;)17 b(!)2170 3862 y Fh(1)p 2168 3874 V 2168 3915 a Fk(g)2214 3864 y Fx(\()p Ft(p)22 b Fx(+)g Ft(q)k Fx(+)c Ft(k)j Fx(+)d(1\))2849 3784 y Fj(\001)2895 3864 y Ft(:)38 4099 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(a)28 b(<)f(b)h Fr(\024)g Ft(a)23 b Fx(+)f(2)1088 4063 y Fv(\000)p Fu(m)1209 4099 y Fx(.)43 b(Then)385 4228 y Fj(\014)385 4288 y(\014)385 4348 y(\014)488 4275 y Fx(1)p 428 4320 168 4 v 428 4411 a Ft(g)t Fx(\()p Ft(b)p Fx(\))628 4343 y Fr(\000)802 4275 y Fx(1)p 738 4320 178 4 v 738 4411 a Ft(g)t Fx(\()p Ft(a)p Fx(\))947 4343 y Fr(\000)1065 4275 y Ft(g)1116 4239 y Fv(0)1138 4275 y Fx(\()p Ft(a)p Fx(\))p 1057 4320 217 4 v 1057 4411 a Ft(g)t Fx(\()p Ft(a)p Fx(\))1235 4382 y Fm(2)1284 4343 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))1574 4228 y Fj(\014)1574 4288 y(\014)1574 4348 y(\014)413 4615 y Fx(=)517 4501 y Fj(\014)517 4560 y(\014)517 4620 y(\014)708 4548 y Fx(1)p 560 4592 345 4 v 560 4684 a Ft(g)t Fx(\()p Ft(b)p Fx(\))p Ft(g)t Fx(\()p Ft(a)p Fx(\))915 4535 y Fj(\000)960 4615 y Ft(g)t Fx(\()p Ft(b)p Fx(\))f Fr(\000)h Ft(g)t Fx(\()p Ft(a)p Fx(\))e Fr(\000)i Ft(g)1600 4574 y Fv(0)1622 4615 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)g Fr(\000)g Ft(a)p Fx(\))2040 4535 y Fj(\001)2108 4615 y Fx(+)2216 4548 y Ft(g)2267 4512 y Fv(0)2289 4548 y Fx(\()p Ft(a)p Fx(\))p 2216 4592 201 4 v 2228 4684 a Ft(g)t Fx(\()p Ft(a)p Fx(\))2426 4615 y(\()p Ft(b)g Fr(\000)f Ft(a)p Fx(\))2716 4505 y Fj(\020)2845 4548 y Fx(1)p 2786 4592 168 4 v 2786 4684 a Ft(g)t Fx(\()p Ft(b)p Fx(\))2986 4615 y Fr(\000)3160 4548 y Fx(1)p 3095 4592 178 4 v 3095 4684 a Ft(g)t Fx(\()p Ft(a)p Fx(\))3283 4505 y Fj(\021)3342 4501 y(\014)3342 4560 y(\014)3342 4620 y(\014)413 4833 y Fr(\024)28 b Fx(2)567 4792 y Fu(p)629 4833 y Fr(\001)22 b Fx(2)728 4792 y Fu(p)789 4833 y Fr(\001)g Fx(2)888 4792 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)p Fu(p)p Fv(\000)p Fm(1)1223 4833 y Fr(\001)g Fx(\()p Ft(b)h Fr(\000)f Ft(a)p Fx(\))h(+)f(2)1733 4792 y Fu(p)1794 4833 y Fr(\001)g Fx(2)1893 4792 y Fu(q)1953 4833 y Fr(\001)g Fx(\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))f Fr(\001)g Fx(2)2414 4792 y Fv(\000)p Fu(p)p Fv(\000)p Fu(q)r Fv(\000)p Fu(k)r Fv(\000)p Fm(1)413 4997 y Fr(\024)28 b Fx(2)567 4956 y Fv(\000)p Fu(k)664 4997 y Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))38 5192 y(for)32 b Ft(m)c Fr(\025)g Ft(\016)448 5207 y Fu(g)488 5192 y Fx(\()p Ft(k)d Fx(+)d(2)p Ft(p)g Fx(+)g(1\))p Ft(;)17 b(!)1121 5190 y Fh(1)p 1119 5202 33 3 v 1119 5243 a Fk(g)1166 5192 y Fx(\()p Ft(p)22 b Fx(+)g Ft(q)k Fx(+)c Ft(k)j Fx(+)d(1\).)p 3682 5133 41 4 v 3682 5189 4 56 v 3720 5189 V 3682 5192 41 4 v 38 5426 a Fp(Theorem)40 b(4.6)g Fx(\(Chain)34 b(Rule\))p Fp(.)43 b Fw(L)-5 b(et)37 b Ft(f)22 b Fx(:)34 b Ft(I)40 b Fr(!)31 b Ft(J)46 b Fw(and)37 b Ft(g)15 b Fx(:)33 b Ft(J)41 b Fr(!)31 b Fn(R)48 b Fw(b)-5 b(e)37 b(c)-5 b(ontinuous)37 b(with)g(deriva-)38 5542 y(tives)d Ft(f)323 5506 y Fv(0)346 5542 y Ft(;)17 b(g)441 5506 y Fv(0)499 5542 y Fw(of)34 b(mo)-5 b(duli)34 b Ft(\016)972 5557 y Fu(f)1018 5542 y Ft(;)17 b(\016)1105 5557 y Fu(g)1145 5542 y Fw(.)45 b(Then)1435 5732 y Fx(\()p Ft(g)26 b Fr(\016)c Ft(f)11 b Fx(\))1715 5690 y Fv(0)1765 5732 y Fx(=)28 b(\()p Ft(g)1958 5690 y Fv(0)2003 5732 y Fr(\016)22 b Ft(f)11 b Fx(\))21 b Fr(\001)h Ft(f)2302 5690 y Fv(0)38 5921 y Fw(is)34 b(a)h(derivative)f(of)h Ft(g)25 b Fr(\016)d Ft(f)46 b Fw(with)34 b(mo)-5 b(dulus)946 6111 y Ft(\016)989 6126 y Fu(g)r Fv(\016)p Fu(f)1106 6111 y Fx(\()p Ft(k)s Fx(\))28 b(=)f(max)1549 6030 y Fj(\000)1594 6111 y Ft(\016)1637 6126 y Fu(g)1678 6111 y Fx(\()p Ft(k)e Fx(+)d(1)g(+)g Ft(r)s Fx(\))p Ft(;)17 b(\016)2231 6126 y Fu(f)2276 6111 y Fx(\()p Ft(k)25 b Fx(+)d(1)g(+)g Ft(q)t Fx(\))2742 6030 y Fj(\001)2787 6111 y Ft(;)38 6300 y Fw(wher)-5 b(e)34 b Fx(2)362 6264 y Fu(r)400 6300 y Ft(;)17 b Fx(2)493 6264 y Fu(q)565 6300 y Fw(ar)-5 b(e)35 b(upp)-5 b(er)35 b(b)-5 b(ounds)34 b(for)g Ft(f)1534 6264 y Fv(0)1557 6300 y Ft(;)17 b(g)1652 6264 y Fv(0)1710 6300 y Fw(in)34 b Ft(I)8 b(;)17 b(J)9 b Fw(,)35 b(r)-5 b(esp)g(e)g(ctively.)p eop %%Page: 14 14 14 13 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(a)28 b(<)f(b)h Fr(\024)g Ft(a)23 b Fx(+)f(2)1088 414 y Fv(\000)p Fu(m)1209 450 y Fx(.)43 b(Then,)34 b(using)f(Lemma)e (4.2)687 540 y Fj(\014)687 600 y(\014)720 625 y Ft(g)t Fx(\()p Ft(f)11 b Fx(\()p Ft(b)p Fx(\)\))22 b Fr(\000)g Ft(g)t Fx(\()p Ft(f)11 b Fx(\()p Ft(a)p Fx(\)\))21 b Fr(\000)i Ft(g)1629 584 y Fv(0)1652 625 y Fx(\()p Ft(f)11 b Fx(\()p Ft(a)p Fx(\)\))p Ft(f)1973 584 y Fv(0)1995 625 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))2413 540 y Fj(\014)2413 600 y(\014)715 786 y Fr(\024)820 701 y Fj(\014)820 761 y(\014)853 786 y Ft(g)t Fx(\()p Ft(f)11 b Fx(\()p Ft(b)p Fx(\)\))22 b Fr(\000)g Ft(g)t Fx(\()p Ft(f)11 b Fx(\()p Ft(a)p Fx(\)\))21 b Fr(\000)i Ft(g)1762 745 y Fv(0)1785 786 y Fx(\()p Ft(f)11 b Fx(\()p Ft(a)p Fx(\)\))2047 705 y Fj(\000)2092 786 y Ft(f)g Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))2576 705 y Fj(\001)2621 701 y(\014)2621 761 y(\014)2676 786 y Fx(+)882 862 y Fj(\014)882 922 y(\014)915 947 y Ft(g)966 906 y Fv(0)989 947 y Fx(\()p Ft(f)g Fx(\()p Ft(a)p Fx(\)\))1251 862 y Fj(\014)1251 922 y(\014)1306 947 y Fr(\001)1356 862 y Fj(\014)1356 922 y(\014)1389 947 y Ft(f)g Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(f)2054 906 y Fv(0)2077 947 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))2494 862 y Fj(\014)2494 922 y(\014)715 1110 y Fr(\024)28 b Fx(2)869 1069 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)p Fv(\000)p Fu(r)1145 1026 y Fj(\014)1145 1086 y(\014)1178 1110 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))23 b Fr(\000)f Ft(f)11 b Fx(\()p Ft(a)p Fx(\))1662 1026 y Fj(\014)1662 1086 y(\014)1717 1110 y Fx(+)22 b(2)1864 1069 y Fu(q)1902 1026 y Fj(\014)1902 1086 y(\014)1935 1110 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))23 b Fr(\000)f Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(f)2600 1069 y Fv(0)2623 1110 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))3040 1026 y Fj(\014)3040 1086 y(\014)715 1274 y Fr(\024)28 b Fx(2)869 1233 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1057 1274 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))f(+)g(2)1516 1233 y Fu(q)1554 1189 y Fj(\014)1554 1249 y(\014)1587 1274 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)g Ft(f)2251 1233 y Fv(0)2274 1274 y Fx(\()p Ft(a)p Fx(\)\()p Ft(b)h Fr(\000)g Ft(a)p Fx(\))2692 1189 y Fj(\014)2692 1249 y(\014)715 1438 y Fr(\024)28 b Fx(2)869 1397 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1057 1438 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))f(+)g(2)1516 1397 y Fu(q)1554 1438 y Fx(2)1603 1397 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)p Fv(\000)p Fu(q)1879 1438 y Fx(\()p Ft(b)h Fr(\000)f Ft(a)p Fx(\))715 1601 y(=)27 b(2)867 1560 y Fv(\000)p Fu(k)965 1601 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))38 1786 y(for)32 b Ft(m)c Fr(\025)g Ft(\016)448 1801 y Fu(g)488 1786 y Fx(\()p Ft(k)d Fx(+)d(1)g(+)g Ft(r)s Fx(\))p Ft(;)17 b(\016)1041 1801 y Fu(f)1086 1786 y Fx(\()p Ft(k)25 b Fx(+)d(1)g(+)g Ft(q)t Fx(\).)p 3682 1727 41 4 v 3682 1782 4 56 v 3720 1782 V 3682 1785 41 4 v 38 1978 a Fp(Lemma)43 b(4.7)h Fx(\(Rolle\))p Fp(.)f Fw(L)-5 b(et)40 b Ft(f)22 b Fx(:)35 b([)p Ft(a;)17 b(b)p Fx(])38 b Fr(!)f Fn(R)50 b Fw(b)-5 b(e)40 b(c)-5 b(ontinuous)40 b(with)g(derivative)f Ft(f)3128 1942 y Fv(0)3151 1978 y Fw(,)i(and)e(assume)38 2094 y Ft(f)11 b Fx(\()p Ft(a)p Fx(\))27 b(=)h Ft(f)11 b Fx(\()p Ft(b)p Fx(\))p Fw(.)45 b(Then)34 b(for)g(every)h Ft(k)c Fr(2)d Fn(N)50 b Fw(we)34 b(c)-5 b(an)35 b(\014nd)f Ft(c)27 b Fr(2)i Fx([)p Ft(a;)17 b(b)p Fx(])35 b Fw(such)g(that)g Fr(j)p Ft(f)2971 2058 y Fv(0)2994 2094 y Fx(\()p Ft(c)p Fx(\))p Fr(j)27 b(\024)h Fx(2)3321 2058 y Fv(\000)p Fu(k)3418 2094 y Fw(.)38 2287 y(Pr)-5 b(o)g(of.)41 b Fx(Let)23 b Ft(\016)544 2302 y Fu(f)613 2287 y Fx(b)s(e)g(the)h(mo)s(dulus)e(of)h (di\013eren)m(tiabilit)m(y)d(of)j Ft(f)11 b Fx(,)25 b(and)e(let)f Ft(a)28 b Fx(=)g Ft(a)2812 2302 y Fm(0)2879 2287 y Ft(<)g(a)3034 2302 y Fm(1)3101 2287 y Ft(<)g Fr(\001)17 b(\001)g(\001)25 b Ft(<)j(a)3503 2302 y Fu(n)3578 2287 y Fx(=)f Ft(b)38 2409 y Fx(suc)m(h)h(that)e Ft(a)508 2424 y Fu(i)p Fm(+1)654 2409 y Fr(\024)j Ft(a)811 2424 y Fu(i)849 2409 y Fx(+)10 b(2)984 2373 y Fv(\000)p Fu(\016)1070 2385 y Fk(f)1109 2373 y Fm(\()p Fu(k)r Fm(+2\))1297 2409 y Fx(.)41 b(Compare)27 b(all)d Fr(j)p Ft(f)1993 2373 y Fv(0)2016 2409 y Fx(\()p Ft(a)2105 2424 y Fu(i)2133 2409 y Fx(\))p Fr(j)i Fx(with)g(2)2490 2373 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2706 2409 y Ft(<)h Fx(2)2858 2373 y Fv(\000)p Fu(k)2955 2409 y Fx(.)42 b(If)26 b(some)h(is)f Ft(<)h Fx(2)3598 2373 y Fv(\000)p Fu(k)3696 2409 y Fx(,)38 2525 y(w)m(e)33 b(are)g(done.)44 b(Otherwise)33 b(w)m(e)h(argue)e(as)h(in)f(Lemma)f(4.2:)455 2690 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))482 2944 y(=)591 2819 y Fu(n)p Fv(\000)p Fm(1)586 2849 y Fj(X)601 3059 y Fu(i)p Fm(=0)730 2863 y Fj(\000)776 2944 y Ft(f)g Fx(\()p Ft(a)924 2959 y Fu(i)p Fm(+1)1042 2944 y Fx(\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(a)1349 2959 y Fu(i)1377 2944 y Fx(\))1415 2863 y Fj(\001)482 3283 y Fx(=)591 3159 y Fu(n)p Fv(\000)p Fm(1)586 3189 y Fj(X)601 3399 y Fu(i)p Fm(=0)730 3203 y Fj(\000)776 3283 y Ft(f)835 3242 y Fv(0)858 3283 y Fx(\()p Ft(a)947 3298 y Fu(i)975 3283 y Fx(\)\()p Ft(a)1102 3298 y Fu(i)p Fm(+1)1242 3283 y Fr(\000)23 b Ft(a)1393 3298 y Fu(i)1421 3283 y Fx(\))1459 3203 y Fj(\001)1527 3283 y Fx(+)1631 3159 y Fu(n)p Fv(\000)p Fm(1)1625 3189 y Fj(X)1640 3399 y Fu(i)p Fm(=0)1769 3203 y Fj(\000)1815 3283 y Ft(f)11 b Fx(\()p Ft(a)1963 3298 y Fu(i)p Fm(+1)2081 3283 y Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)2389 3298 y Fu(i)2417 3283 y Fx(\))22 b Fr(\000)g Ft(f)2635 3242 y Fv(0)2658 3283 y Fx(\()p Ft(a)2747 3298 y Fu(i)2776 3283 y Fx(\)\()p Ft(a)2903 3298 y Fu(i)p Fm(+1)3043 3283 y Fr(\000)h Ft(a)3194 3298 y Fu(i)3222 3283 y Fx(\))3260 3203 y Fj(\001)482 3533 y Fr(\025)29 b Fx(2)637 3492 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)824 3533 y Fx(\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\))h Fr(\000)f Fx(2)1285 3492 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)1473 3533 y Fx(\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))28 b Ft(>)f Fx(0)p Ft(:)38 3712 y Fx(This)33 b(con)m(tradiction)e (pro)m(v)m(es)j(the)f(claim.)p 3682 3653 V 3682 3709 4 56 v 3720 3709 V 3682 3712 41 4 v 38 3905 a Fp(Theorem)h(4.8)h Fx(\(Mean)c(V)-8 b(alue)30 b(Theorem\))p Fp(.)40 b Fw(L)-5 b(et)33 b Ft(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)f Fn(R)44 b Fw(b)-5 b(e)33 b(c)-5 b(ontinuous)32 b(with)h(derivative)38 4021 y Ft(f)97 3985 y Fv(0)120 4021 y Fw(.)45 b(Then)34 b(for)g(every)h Ft(k)c Fr(2)d Fn(N)50 b Fw(we)34 b(c)-5 b(an)34 b(\014nd)h Ft(c)27 b Fr(2)i Fx([)p Ft(a;)17 b(b)p Fx(])35 b Fw(such)g(that)1001 4119 y Fj(\014)1001 4179 y(\014)1034 4204 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(f)1699 4162 y Fv(0)1722 4204 y Fx(\()p Ft(c)p Fx(\)\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))2130 4119 y Fj(\014)2130 4179 y(\014)2191 4204 y Fr(\024)28 b Fx(2)2345 4162 y Fv(\000)p Fu(k)2442 4204 y Fx(\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\))p Ft(:)38 4405 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b(2)560 4369 y Fv(\000)p Fu(l)668 4405 y Fr(\024)28 b Fx(2)822 4369 y Fv(\000)p Fu(k)920 4405 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))32 b(and)h(de\014ne)h(a)e(con)m(tin)m(uous)h Ft(h)11 b Fx(:)34 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)g Fn(R)43 b Fx(b)m(y)953 4585 y Ft(h)p Fx(\()p Ft(x)p Fx(\))28 b(:=)g(\()p Ft(x)22 b Fr(\000)h Ft(a)p Fx(\))1603 4505 y Fj(\000)1649 4585 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))2133 4505 y Fj(\001)2200 4585 y Fr(\000)23 b Ft(f)11 b Fx(\()p Ft(x)p Fx(\)\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\))p Ft(:)38 4765 y Fx(Then)29 b Ft(h)p Fx(\()p Ft(a)p Fx(\))f(=)g Ft(h)p Fx(\()p Ft(b)p Fx(\))g(=)f Fr(\000)p Ft(f)11 b Fx(\()p Ft(a)p Fx(\)\()p Ft(b)i Fr(\000)g Ft(a)p Fx(\).)44 b(Hence)30 b(b)m(y)f(Lemma)e(4.7)g(w)m(e)j(can)e(\014nd)h Ft(c)f Fx(in)g([)p Ft(a;)17 b(b)p Fx(])28 b(suc)m(h)i(that)957 4864 y Fj(\014)957 4923 y(\014)990 4948 y Ft(h)1046 4907 y Fv(0)1070 4948 y Fx(\()p Ft(c)p Fx(\))1188 4864 y Fj(\014)1188 4923 y(\014)1248 4948 y Fx(=)1352 4864 y Fj(\014)1352 4923 y(\014)1385 4948 y Ft(f)11 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(f)2050 4907 y Fv(0)2073 4948 y Fx(\()p Ft(c)p Fx(\)\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))2481 4864 y Fj(\014)2481 4923 y(\014)2542 4948 y Fr(\024)28 b Fx(2)2696 4907 y Fv(\000)p Fu(l)2776 4948 y Ft(:)38 5127 y Fx(This)33 b(pro)m(v)m(es)h(the)f(claim.)p 3682 5068 V 3682 5123 4 56 v 3720 5123 V 3682 5126 41 4 v 1514 5386 a(5.)49 b Fs(Integra)-7 b(tion)137 5560 y Fx(F)f(or)29 b(simplicit)m(y)e(w)m(e) k(de\014ne)1171 5480 y Fj(R)1237 5506 y Fu(b)1218 5595 y(a)1288 5560 y Ft(f)11 b Fx(\()p Ft(a)p Fx(\))17 b Ft(dx)29 b Fx(for)h(rational)d(end)j(p)s(oin)m(ts)f Ft(a;)17 b(b)30 b Fx(only)-8 b(.)42 b(F)-8 b(or)29 b(reals)h Ft(a;)17 b(b)30 b Fx(one)38 5703 y(can)j(easily)e(de\014ne)767 5623 y Fj(R)834 5649 y Fu(b)814 5738 y(a)885 5703 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)32 b Fx(b)m(y)h(rational)d(appro)m (ximation.)38 5851 y Fp(De\014nition)45 b(5.1.)g Fx(Let)c Ft(a;)17 b(b)40 b Fx(b)s(e)g(rationals)f(with)g Ft(a)i(<)g(b)p Fx(.)66 b(A)41 b(list)d Ft(P)54 b Fx(=)41 b Ft(a)2839 5866 y Fm(0)2878 5851 y Ft(;)17 b(:)g(:)g(:)f(;)h(a)3148 5866 y Fu(n)3235 5851 y Fx(of)40 b(rationals)38 5967 y(is)f(a)h Fw(p)-5 b(artition)48 b Fx(of)39 b(the)i(in)m(terv)-5 b(al)39 b([)p Ft(a;)17 b(b)p Fx(],)42 b(if)d Ft(a)i Fx(=)f Ft(a)1914 5982 y Fm(0)1994 5967 y Fr(\024)h Ft(a)2163 5982 y Fm(1)2243 5967 y Fr(\024)g(\001)17 b(\001)g(\001)39 b(\024)i Ft(a)2687 5982 y Fu(n)2775 5967 y Fx(=)f Ft(b)p Fx(.)67 b(max)o Fr(f)17 b Ft(a)3325 5982 y Fu(i)p Fm(+1)3471 5967 y Fr(\000)27 b Ft(a)3626 5982 y Fu(i)3695 5967 y Fr(j)38 6083 y Ft(i)41 b(<)g(n)17 b Fr(g)40 b Fx(is)g(the)h Fw(mesh)47 b Fx(of)40 b Ft(P)14 b Fx(.)67 b(A)40 b(partition)f Ft(Q)i Fx(=)g Ft(a)2042 6047 y Fv(0)2042 6108 y Fm(0)2082 6083 y Ft(;)17 b(:)g(:)g(:)e(;)i(a)2351 6047 y Fv(0)2351 6108 y Fu(m)2458 6083 y Fx(of)40 b([)p Ft(a;)17 b(b)p Fx(])41 b(is)f(a)g Fw(r)-5 b(e\014nement)49 b Fx(of)40 b Ft(P)14 b Fx(,)38 6199 y(if)37 b Fr(8)p Ft(i)i Fr(\024)f Ft(n)p Fr(9)p Ft(j)45 b Fr(\024)39 b Ft(m)p Fx(\()p Ft(a)863 6163 y Fv(0)863 6224 y Fu(j)938 6199 y Fx(=)f Ft(a)1103 6214 y Fu(i)1131 6199 y Fx(\).)62 b(If)38 b Ft(f)22 b Fx(:)35 b([)p Ft(a;)17 b(b)p Fx(])39 b Fr(!)f Fn(R)49 b Fx(is)38 b(a)h(con)m(tin)m(uous)g(function)f(giv)m(en)h(b)m(y)h Ft(h)3503 6214 y Fu(f)3548 6199 y Fx(,)g Ft(\013)3677 6214 y Fu(f)38 6320 y Fx(and)35 b Ft(!)291 6335 y Fu(f)337 6320 y Fx(,)h(and)g Ft(P)46 b Fx(=)32 b Ft(a)861 6335 y Fm(0)901 6320 y Ft(;)17 b(a)996 6335 y Fm(1)1035 6320 y Ft(;)g(:)g(:)g(:)f(;)h(a)1305 6335 y Fu(n)1388 6320 y Fx(a)35 b(partition)e(of)i([)p Ft(a;)17 b(b)p Fx(],)37 b(then)f(an)g(arbitrary)e(sum)i(of)f(the)h(form)p eop %%Page: 15 15 15 14 bop 38 384 a Fj(P)143 410 y Fu(n)p Fv(\000)p Fm(1)143 487 y Fu(i)p Fm(=0)297 458 y Ft(h)353 473 y Fu(f)398 458 y Fx(\()p Ft(e)481 473 y Fu(i)510 458 y Ft(;)17 b(n)p Fx(\))24 b Fr(\001)f Fx(\()p Ft(a)814 473 y Fu(i)p Fm(+1)957 458 y Fr(\000)i Ft(a)1110 473 y Fu(i)1138 458 y Fx(\))35 b(with)g Ft(e)1481 473 y Fu(i)1542 458 y Fr(2)e Fx([)p Ft(a)1719 473 y Fu(i)1748 458 y Ft(;)17 b(a)1843 473 y Fu(i)p Fm(+1)1961 458 y Fx(])36 b(is)f(denoted)h(b)m(y)h Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\).)51 b(In)36 b(particular)d(for)38 579 y Ft(a)89 594 y Fu(i)145 579 y Fx(=)27 b Ft(a)c Fx(+)439 540 y Fu(i)p 430 556 43 4 v 430 613 a(n)483 579 y Fx(\()p Ft(b)f Fr(\000)h Ft(a)p Fx(\))1178 812 y Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\))27 b(:=)1823 745 y Ft(b)22 b Fr(\000)h Ft(a)p 1823 789 215 4 v 1901 880 a(n)2069 687 y Fu(n)p Fv(\000)p Fm(1)2064 717 y Fj(X)2079 927 y Fu(i)p Fm(=0)2224 812 y Ft(h)2280 827 y Fu(f)2326 812 y Fx(\()p Ft(a)2415 827 y Fu(i)2443 812 y Ft(;)17 b(n)p Fx(\))38 1054 y(is)32 b(one)h(of)f(the)h(n)m(um)m(b)s(ers)g Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\).)38 1197 y Fp(Theorem)36 b(5.2.)41 b Fw(Assume)34 b(that)g Ft(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])29 b Fr(!)e Fn(R)45 b Fw(is)34 b(c)-5 b(ontinuous)33 b(with)h(mo)-5 b(dulus)34 b Ft(!)3113 1212 y Fu(f)3192 1197 y Fw(of)g(\(uniform\))38 1313 y(c)-5 b(ontinuity.)76 b(Then)44 b Fx(\()p Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\)\))1384 1328 y Fu(n)p Fv(2)p Fo(N)1571 1313 y Fw(is)45 b(a)53 b Fs(Ca)n(uchy)45 b Fw(se)-5 b(quenc)g(e)45 b(of)g(r)-5 b(ationals)44 b(with)h(mo)-5 b(dulus)38 1432 y Ft(\013)q Fx(\()p Ft(p)p Fx(\))27 b(=)h(2)406 1396 y Fu(!)450 1408 y Fk(f)489 1396 y Fm(\()p Fu(p)p Fm(+)p Fu(q)r Fm(+1\))762 1432 y Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))p Fw(,)35 b(wher)-5 b(e)34 b Ft(q)k Fw(is)d(such)g(that)g Ft(b)23 b Fr(\000)f Ft(a)28 b Fr(\024)g Fx(2)2396 1396 y Fu(q)2434 1432 y Fw(;)35 b(we)f(denote)h(this)f(r)-5 b(e)g(al)35 b(by)1635 1539 y Fj(Z)1735 1565 y Fu(b)1690 1765 y(a)1786 1675 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx:)38 1913 y Fw(Mor)-5 b(e)g(over,)34 b(if)h Ft(P)48 b Fw(is)35 b(a)f(p)-5 b(artition)35 b(of)g(mesh)f Fr(\024)28 b Fx(2)1809 1877 y Fv(\000)p Fu(!)1908 1889 y Fk(f)1947 1877 y Fm(\()p Fu(l)q Fm(\))2028 1913 y Fw(,)35 b(then)1107 2035 y Fj(\014)1107 2095 y(\014)1107 2155 y(\014)1140 2150 y Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\))21 b Fr(\000)1577 2014 y Fj(Z)1677 2040 y Fu(b)1632 2240 y(a)1728 2150 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)2041 2035 y Fj(\014)2041 2095 y(\014)2041 2155 y(\014)2101 2150 y Fr(\024)28 b Fx(2)2255 2109 y Fv(\000)p Fu(l)2336 2150 y Fx(\()p Ft(b)23 b Fr(\000)f Ft(a)p Fx(\))p Ft(:)38 2389 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)25 b Ft(k)s Fx(,)i Ft(l)h Fx(b)s(e)d(giv)m(en)h(and)f Ft(P)41 b Fx(=)28 b Ft(a)1482 2404 y Fm(0)1522 2389 y Ft(;)17 b(:)g(:)g(:)e(;)i(a)1791 2404 y Fu(n)1838 2389 y Fx(,)27 b Ft(Q)h Fx(=)f Ft(b)2141 2404 y Fm(0)2181 2389 y Ft(;)17 b(:)g(:)g(:)f(;)h(b)2441 2404 y Fu(m)2533 2389 y Fx(partitions)24 b(of)h([)p Ft(a;)17 b(b)p Fx(])26 b(with)f(mesh)38 2508 y Fr(\024)37 b Fx(2)201 2472 y Fv(\000)p Fu(!)300 2484 y Fk(f)340 2472 y Fm(\()p Fu(k)r Fm(\))475 2508 y Fx(or)h Fr(\024)g Fx(2)764 2472 y Fv(\000)p Fu(!)863 2484 y Fk(f)902 2472 y Fm(\()p Fu(l)q Fm(\))983 2508 y Fx(,)i(resp)s(ectiv)m(ely)-8 b(.)61 b(Let)38 b Ft(R)g Fx(=)f Ft(c)2075 2523 y Fm(0)2115 2508 y Ft(;)17 b(:)g(:)g(:)f(;)h(c)2376 2523 y Fu(r)2451 2508 y Fx(b)s(e)39 b(the)f(common)f(re\014nemen)m(t)i(of)38 2624 y Ft(P)49 b Fx(and)35 b Ft(Q)p Fx(,)i(obtained)e(b)m(y)h(arranging)e Ft(a)1518 2639 y Fm(0)1558 2624 y Ft(;)17 b(:)g(:)g(:)f(;)h(a)1828 2639 y Fu(n)1874 2624 y Ft(;)g(b)1959 2639 y Fm(0)1999 2624 y Ft(;)g(:)g(:)g(:)f(;)h(b)2259 2639 y Fu(m)2361 2624 y Fx(in)m(to)35 b(a)g(monotone)g(sequence)j(\(here)38 2740 y(w)m(e)43 b(mak)m(e)f(use)h(of)e(the)i(fact)e(that)h Ft(a)1404 2755 y Fu(i)1433 2740 y Fx(,)i Ft(b)1545 2755 y Fu(j)1624 2740 y Fx(are)e(assumed)g(to)g(b)s(e)g(rational\).)69 b(Let)42 b Ft(d)3169 2755 y Fu(k)3255 2740 y Fr(2)i Fx([)p Ft(c)3434 2755 y Fu(k)3477 2740 y Ft(;)17 b(c)3563 2755 y Fu(k)r Fm(+1)3696 2740 y Fx(])38 2857 y(for)36 b Ft(k)j(<)c(r)s Fx(.)57 b(F)-8 b(or)36 b(ev)m(ery)j Ft(i)d(<)f(n)i Fx(denote)h(b)m(y) 1698 2782 y Fj(P)1803 2886 y Fu(i)1869 2857 y Fx(the)f(summation)e(o)m (v)m(er)j(all)d(indices)i Ft(k)j Fx(suc)m(h)f(that)38 2973 y Ft(a)89 2988 y Fu(i)145 2973 y Fr(\024)28 b Ft(c)292 2988 y Fu(k)362 2973 y Ft(<)g(a)517 2988 y Fu(i)p Fm(+1)635 2973 y Fx(.)44 b(Then)452 3132 y Fr(j)p Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\))21 b Fr(\000)i Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(R)q Fx(\))p Fr(j)480 3385 y Fx(=)583 3271 y Fj(\014)583 3330 y(\014)583 3390 y(\014)622 3261 y Fu(n)p Fv(\000)p Fm(1)617 3291 y Fj(X)631 3501 y Fu(i)p Fm(=0)777 3385 y Ft(h)833 3400 y Fu(f)879 3385 y Fx(\()p Ft(e)962 3400 y Fu(i)990 3385 y Ft(;)g(n)p Fx(\))22 b Fr(\001)g Fx(\()p Ft(a)1291 3400 y Fu(i)p Fm(+1)1431 3385 y Fr(\000)h Ft(a)1582 3400 y Fu(i)1610 3385 y Fx(\))g Fr(\000)1780 3261 y Fu(r)r Fv(\000)p Fm(1)1770 3291 y Fj(X)1778 3503 y Fu(k)r Fm(=0)1931 3385 y Ft(h)1987 3400 y Fu(f)2032 3385 y Fx(\()p Ft(d)2121 3400 y Fu(k)2163 3385 y Ft(;)17 b(r)s Fx(\))22 b Fr(\001)g Fx(\()p Ft(c)2444 3400 y Fu(k)r Fm(+1)2599 3385 y Fr(\000)g Ft(c)2740 3400 y Fu(k)2783 3385 y 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4082 y Fu(f)1035 4067 y Fx(\()p Ft(e)1118 4082 y Fu(i)1147 4067 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)1464 4082 y Fu(f)1510 4067 y Fx(\()p Ft(d)1599 4082 y Fu(k)1641 4067 y Ft(;)17 b(r)s Fx(\))p Fr(j)p Fx(\()p Ft(c)1878 4082 y Fu(k)r Fm(+1)2032 4067 y Fr(\000)23 b Ft(c)2174 4082 y Fu(k)2216 4067 y Fx(\))480 4407 y Fr(\024)590 4283 y Fu(n)p Fv(\000)p Fm(1)585 4313 y Fj(X)600 4523 y Fu(i)p Fm(=0)746 4313 y Fj(X)805 4523 y Fu(i)889 4327 y Fj(\000)935 4407 y Fr(j)p Ft(h)1019 4422 y Fu(f)1064 4407 y Fx(\()p Ft(e)1147 4422 y Fu(i)1176 4407 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)1493 4422 y Fu(f)1539 4407 y Fx(\()p Ft(e)1622 4422 y Fu(i)1650 4407 y Ft(;)17 b(r)s Fx(\))p Fr(j)k Fx(+)h Fr(j)p Ft(h)2010 4422 y Fu(f)2055 4407 y Fx(\()p Ft(e)2138 4422 y Fu(i)2167 4407 y Ft(;)17 b(r)s Fx(\))k Fr(\000)i Ft(h)2473 4422 y Fu(f)2518 4407 y Fx(\()p Ft(d)2607 4422 y Fu(k)2650 4407 y Ft(;)17 b(r)s Fx(\))p Fr(j)2807 4327 y Fj(\001)2851 4407 y Fx(\()p Ft(c)2931 4422 y Fu(k)r Fm(+1)3086 4407 y 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6167 y Fm(\()p Fu(p)p Fm(+)p Fu(q)r Fm(+1\))444 6204 y Fx(\()p Ft(b)14 b Fr(\000)g Ft(a)p Fx(\))p Ft(;)j(\013)823 6219 y Fu(f)869 6204 y Fx(\()p Ft(p)d Fx(+)g Ft(q)19 b Fx(+)14 b(2\),)29 b(where)h Ft(q)j Fx(is)28 b(suc)m(h)i(that)f Ft(b)14 b Fr(\000)g Ft(a)29 b Fr(\024)f Fx(2)2607 6167 y Fu(q)2645 6204 y Fx(;)i(therefore)3108 6123 y Fj(\000)3153 6204 y Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\))3630 6123 y Fj(\001)3676 6243 y Fu(n)38 6320 y Fx(is)32 b(a)g Fs(Ca)n(uchy)j Fx(sequence.)p eop %%Page: 16 16 16 15 bop 137 453 a Fx(Moreo)m(v)m(er)35 b(w)m(e)e(ha)m(v)m(e)h(for)e Ft(n)c Fr(\025)g Fx(2)1323 417 y Fu(!)1367 429 y Fk(f)1407 417 y Fm(\()p Fu(l)q Fm(\))1487 453 y Fx(\()p Ft(b)23 b Fr(\000)g Ft(a)p Fx(\))p Ft(;)17 b(\013)1884 468 y Fu(f)1929 453 y Fx(\()p Ft(l)24 b Fx(+)e(1\))935 709 y Fr(j)p Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\))20 b Fr(\000)j Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\))p Fr(j)27 b(\024)i Fx(\(2)2124 668 y Fv(\000)p Fu(k)2243 709 y Fx(+)22 b(2)2390 668 y Fv(\000)p Fu(l)2471 709 y Fx(\)\()p Ft(b)g Fr(\000)h Ft(a)p Fx(\))p Ft(:)38 957 y Fx(No)m(w)33 b(let)f Ft(n)c Fr(!)f(1)32 b Fx(and)h Ft(k)e Fr(!)c(1)p Fx(.)43 b(Then)34 b(w)m(e)f(obtain)1113 1254 y Fr(j)p Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(P)d Fx(\))20 b Fr(\000)1577 1118 y Fj(Z)1677 1144 y Fu(b)1632 1344 y(a)1728 1254 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)p Fr(j)27 b(\024)h Fx(2)2250 1212 y Fv(\000)p Fu(l)2331 1254 y Fx(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\))p Ft(;)38 1553 y Fx(as)33 b(w)m(as)g(to)f(b)s(e)h(sho)m(wn.)p 3682 1494 41 4 v 3682 1550 4 56 v 3720 1550 V 3682 1553 41 4 v 137 1832 a(Up)e(to)f(no)m(w)h(w)m(e)g(ha)m(v)m(e)g(alw)m(a)m(ys)g (considered)g(the)g(in)m(tegral)d(with)i(resp)s(ect)h(to)f(a)g(\014xed) h(in)m(tegration)38 1948 y(in)m(terv)-5 b(al.)84 b(No)m(w)47 b(w)m(e)h(view)e(the)h(upp)s(er)g(b)s(ound)g(of)f(this)g(in)m(terv)-5 b(al)45 b(as)i(v)-5 b(ariable)45 b(and)i(study)g(the)38 2064 y(function)32 b(obtained)g(in)g(this)g(w)m(a)m(y)-8 b(.)45 b(It)32 b(is)g(called)g(the)h(\\undetermined)f(in)m(tegral".)137 2180 y(Giv)m(en)k(a)g(con)m(tin)m(uous)g Ft(f)22 b Fx(:)34 b([)p Ft(a)1208 2195 y Fm(0)1248 2180 y Ft(;)17 b(b)1333 2195 y Fm(0)1373 2180 y Fx(])33 b Fr(!)f Fn(R)5 b Fx(,)43 b(w)m(e)37 b(\014rst)f(need)h(to)e(establish)h Ft(F)14 b Fx(\()p Ft(x)p Fx(\))33 b(:=)3188 2100 y Fj(R)3254 2127 y Fu(x)3235 2215 y(a)3272 2224 y Fh(0)3328 2180 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)35 b Fx(as)38 2302 y(a)e(con)m(tin)m(uous)h(function.)45 b(This)33 b(means)g(that)g(w)m(e)i(ha)m(v)m(e)f(to)f(come)g(up)h(with)e Ft(h)2925 2317 y Fu(F)2984 2302 y Fx(,)i Ft(\013)3107 2317 y Fu(F)3198 2302 y Fx(and)g Ft(!)3450 2317 y Fu(F)3508 2302 y Fx(.)46 b(F)-8 b(or)38 2418 y(simplicit)m(y)30 b(assume)j([)p Ft(a)898 2433 y Fm(0)938 2418 y Ft(;)17 b(b)1023 2433 y Fm(0)1062 2418 y Fx(])28 b(=)f([0)p Ft(;)17 b Fx(1].)43 b(Let)1374 2667 y Ft(h)1430 2682 y Fu(F)1488 2667 y Fx(\()p Ft(a;)17 b(n)p Fx(\))28 b(:=)g Ft(S)6 b Fx(\()p Ft(f)f(;)17 b Fx(0)p Ft(;)g(a;)g(n)p Fx(\))p Ft(:)38 2915 y Fx(By)27 b(Theorem)h(5.2)f(w)m(e)h(already)e(kno)m(w)j (that)e(\()p Ft(h)1770 2930 y Fu(F)1828 2915 y Fx(\()p Ft(a;)17 b(n)p Fx(\)\))2095 2930 y Fu(n)2169 2915 y Fx(is)27 b(a)g Fs(Ca)n(uchy)i Fx(sequence)h(with)d(mo)s(dulus)38 3034 y Ft(k)k Fr(7!)c Fx(2)296 2998 y Fu(!)340 3010 y Fk(f)379 2998 y Fm(\()p Fu(k)r Fm(+1\))567 3034 y Fx(.)43 b(It)29 b(remains)g(to)h(pro)m(vide)g(a)f(mo)s(dulus)g Ft(!)2091 3049 y Fu(F)2179 3034 y Fx(of)g(\(uniform\))f(con)m(tin)m (uit)m(y)-8 b(.)43 b(T)-8 b(o)30 b(this)f(end,)38 3151 y(w)m(e)36 b(divide)e(the)i(in)m(terv)-5 b(als)34 b([0)p Ft(;)17 b(a)p Fx(])35 b(and)g([0)p Ft(;)17 b(b)p Fx(])35 b(in)g Ft(n)g Fx(pieces)h(eac)m(h,)g(and)f(let)g Ft(a)2819 3166 y Fu(i)2879 3151 y Fx(:=)3033 3112 y Fu(i)p 3024 3128 43 4 v 3024 3185 a(n)3077 3151 y Ft(a)g Fx(and)g Ft(b)3396 3166 y Fu(i)3457 3151 y Fx(:=)3611 3112 y Fu(i)p 3601 3128 V 3601 3185 a(n)3654 3151 y Ft(b)p Fx(.)38 3267 y(Then)309 3408 y Fj(\014)309 3468 y(\014)342 3492 y Ft(h)398 3507 y Fu(F)457 3492 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)864 3507 y Fu(F)923 3492 y Fx(\()p Ft(b;)17 b(n)p Fx(\))1142 3408 y Fj(\014)1142 3468 y(\014)1203 3492 y Fx(=)1306 3408 y Fj(\014)1306 3468 y(\014)1340 3492 y Ft(S)6 b Fx(\()p Ft(f)f(;)17 b Fx(0)p Ft(;)g(a;)g(n)p Fx(\))k Fr(\000)i Ft(S)6 b Fx(\()p Ft(f)f(;)17 b Fx(0)p Ft(;)g(b;)g(n)p Fx(\))2421 3408 y Fj(\014)2421 3468 y(\014)1203 3746 y Fx(=)1321 3679 y(1)p 1316 3723 59 4 v 1316 3814 a Ft(n)1385 3661 y Fj(\014)1385 3721 y(\014)1418 3746 y Ft(a)1491 3622 y Fu(n)p Fv(\000)p Fm(1)1486 3651 y Fj(X)1500 3861 y Fu(i)p Fm(=0)1646 3746 y Ft(h)1702 3761 y Fu(f)1748 3746 y Fx(\()p Ft(a)1837 3761 y Fu(i)1865 3746 y Ft(;)g(n)p Fx(\))22 b Fr(\000)h Ft(b)2190 3622 y Fu(n)p Fv(\000)p Fm(1)2185 3651 y Fj(X)2200 3861 y Fu(i)p Fm(=0)2345 3746 y Ft(h)2401 3761 y Fu(f)2447 3746 y Fx(\()p Ft(b)2526 3761 y Fu(i)2554 3746 y Ft(;)17 b(n)p Fx(\))2694 3661 y Fj(\014)2694 3721 y(\014)1203 4086 y Fr(\024)1323 4019 y Fx(1)p 1318 4063 V 1318 4154 a Ft(n)1408 3961 y Fu(n)p Fv(\000)p Fm(1)1403 3991 y Fj(X)1418 4201 y Fu(i)p Fm(=0)1547 4005 y Fj(\000)1592 4086 y Ft(a)1643 4001 y Fj(\014)1643 4061 y(\014)1677 4086 y Ft(h)1733 4101 y Fu(f)1778 4086 y Fx(\()p Ft(a)1867 4101 y Fu(i)1896 4086 y Ft(;)g(n)p Fx(\))22 b Fr(\000)g Ft(h)2213 4101 y Fu(f)2259 4086 y Fx(\()p Ft(b)2338 4101 y Fu(i)2366 4086 y Ft(;)17 b(n)p Fx(\))2506 4001 y Fj(\014)2506 4061 y(\014)2562 4086 y Fx(+)22 b Fr(j)p Ft(a)g Fr(\000)g Ft(b)p Fr(j)h(\001)e(j)p Ft(h)3085 4101 y Fu(f)3131 4086 y Fx(\()p Ft(b)3210 4101 y Fu(i)3238 4086 y Ft(;)c(n)p Fx(\))p Fr(j)3406 4005 y Fj(\001)1203 4426 y Fr(\024)1323 4359 y Fx(1)p 1318 4403 V 1318 4494 a Ft(n)1408 4301 y Fu(n)p Fv(\000)p Fm(1)1403 4331 y Fj(X)1418 4541 y Fu(i)p Fm(=0)1547 4345 y Fj(\000)1592 4426 y Ft(a)23 b Fr(\001)f Fx(2)1765 4385 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1974 4426 y Fx(+)g Fr(j)p Ft(a)g Fr(\000)h Ft(b)p Fr(j)f(\001)g Fx(2)2463 4385 y Fu(p)2502 4345 y Fj(\001)2576 4426 y Fr(\024)28 b Fx(2)2730 4385 y Fv(\000)p Fu(k)38 4778 y Fx(pro)m(vided)35 b Fr(j)p Ft(a)520 4793 y Fu(i)572 4778 y Fr(\000)25 b Ft(b)715 4793 y Fu(i)743 4778 y Fr(j)32 b(\024)h Fx(2)962 4742 y Fv(\000)p Fu(!)1061 4754 y Fk(f)1100 4742 y Fm(\()p Fu(k)r Fm(+1\)+1)1413 4778 y Fx(and)j Fr(j)p Ft(a)23 b Fr(\000)i Ft(b)p Fr(j)32 b(\024)h Fx(2)2070 4742 y Fv(\000)p Fu(p)p Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2348 4778 y Fx(,)j(where)g Ft(p)f Fx(is)g(c)m(hosen)i(according)d(to)38 4894 y(Lemma)d(3.9)h(suc)m(h)i(that)f Ft(h)1031 4909 y Fu(f)1076 4894 y Fx(\()p Ft(b)1155 4909 y Fu(i)1184 4894 y Ft(;)17 b(n)p Fx(\))27 b Fr(\024)h Fx(2)1505 4858 y Fu(p)1545 4894 y Fx(,)k(for)g Ft(n)c Fr(\025)h Ft(\013)2007 4909 y Fu(f)2052 4894 y Fx(\(0\).)43 b(So)33 b(let)481 5154 y Ft(\013)543 5169 y Fu(F)602 5154 y Fx(\()p Ft(k)s Fx(\))27 b(:=)h(max)1072 5073 y Fj(\000)1117 5154 y Ft(\013)1179 5169 y Fu(f)1225 5154 y Fx(\(0\))p Ft(;)17 b Fx(2)1443 5112 y Fu(!)1487 5124 y Fk(f)1525 5112 y Fm(\()p Fu(k)r Fm(+1\))1713 5073 y Fj(\001)1759 5154 y Ft(;)114 b(!)1961 5169 y Fu(F)2019 5154 y Fx(\()p Ft(k)s Fx(\))28 b(:=)g(max)2489 5073 y Fj(\000)2535 5154 y Ft(p)22 b Fx(+)g Ft(k)s(;)17 b(!)2863 5169 y Fu(f)2908 5154 y Fx(\()p Ft(k)25 b Fx(+)d(1\))3207 5073 y Fj(\001)3252 5154 y Ft(:)38 5404 y Fp(Theorem)35 b(5.3.)41 b Fw(L)-5 b(et)34 b Ft(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])29 b Fr(!)e Fn(R)44 b Fw(b)-5 b(e)34 b(c)-5 b(ontinuous)33 b(with)h(mo)-5 b(dulus)33 b Ft(!)2731 5419 y Fu(f)2810 5404 y Fw(of)g(\(uniform\))g(c)-5 b(ontinu-)38 5520 y(ity.)45 b(F)-7 b(or)34 b Ft(x)28 b Fr(2)g Fx([)p Ft(a;)17 b(b)p Fx(])36 b Fw(let)1467 5783 y Ft(F)14 b Fx(\()p Ft(x)p Fx(\))28 b(:=)1834 5647 y Fj(Z)1933 5674 y Fu(x)1889 5873 y(a)1994 5783 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt:)38 6068 y Fw(b)-5 b(e)40 b(the)h(c)-5 b(ontinuous)40 b(function)g(just)h(describ)-5 b(e)g(d.)61 b(Then)40 b(this)g(function)g Ft(F)25 b Fx(:)36 b([)p Ft(a;)17 b(b)p Fx(])38 b Fr(!)g Fn(R)52 b Fw(has)40 b Ft(f)51 b Fw(as)38 6184 y(derivative,)35 b(with)g(mo)-5 b(dulus)36 b Ft(!)1169 6199 y Fu(f)1214 6184 y Fw(.)47 b(Mor)-5 b(over,)36 b(if)g Ft(G)f Fw(is)h(any)f(di\013er)-5 b(entiable)35 b(function)g(on)h Fx([)p Ft(a;)17 b(b)p Fx(])36 b Fw(with)38 6300 y Ft(G)115 6264 y Fv(0)166 6300 y Fx(=)27 b Ft(f)11 b Fw(,)35 b(then)f(the)h(di\013er)-5 b(enc)g(e)34 b Ft(F)i Fr(\000)22 b Ft(G)35 b Fw(is)g(a)f(c)-5 b(onstant)35 b(function.)p eop %%Page: 17 17 17 16 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b Ft(m)28 b Fr(\025)g Ft(!)790 465 y Fu(f)835 450 y Fx(\()p Ft(k)s Fx(\))k(and)h Ft(h)28 b Fx(:=)g Fr(\006)p Fx(2)1528 414 y Fv(\000)p Fu(m)1649 450 y Fx(.)44 b(Then)570 605 y Fj(\014)570 665 y(\014)570 725 y(\014)613 653 y Ft(F)14 b Fx(\()p Ft(y)t Fx(\))21 b Fr(\000)i Ft(F)14 b Fx(\()p Ft(x)p Fx(\))p 613 697 534 4 v 852 788 a Ft(h)1179 720 y Fr(\000)23 b Ft(f)11 b Fx(\()p Ft(x)p Fx(\))1469 605 y Fj(\014)1469 665 y(\014)1469 725 y(\014)1529 720 y Fx(=)1633 605 y Fj(\014)1633 665 y(\014)1633 725 y(\014)1680 653 y Fx(1)p 1676 697 57 4 v 1676 788 a Ft(h)1742 609 y Fj(\020)1802 584 y(Z)1901 611 y Fu(y)1857 810 y(a)1959 720 y Ft(f)g Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)22 b Fr(\000)2354 584 y Fj(Z)2453 611 y Fu(x)2409 810 y(a)2514 720 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)2787 609 y Fj(\021)2868 720 y Fr(\000)22 b Ft(f)11 b Fx(\()p Ft(x)p Fx(\))3157 605 y Fj(\014)3157 665 y(\014)3157 725 y(\014)1529 985 y Fx(=)1633 871 y Fj(\014)1633 930 y(\014)1633 990 y(\014)1680 918 y Fx(1)p 1676 962 V 1676 1054 a Ft(h)1759 850 y Fj(Z)1859 876 y Fu(y)1814 1075 y(x)1917 985 y Ft(f)g Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)k Fr(\000)2324 918 y Fx(1)p 2321 962 V 2321 1054 a Ft(h)2403 850 y Fj(Z)2503 876 y Fu(y)2459 1075 y(x)2561 985 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dt)2854 871 y Fj(\014)2854 930 y(\014)2854 990 y(\014)1529 1250 y Fr(\024)1648 1183 y Fx(1)p 1645 1227 V 1645 1319 a Ft(h)1727 1115 y Fj(Z)1827 1141 y Fu(y)1783 1340 y(x)1885 1250 y Fr(j)p Ft(f)11 b Fx(\()p Ft(t)p Fx(\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(x)p Fx(\))p Fr(j)17 b Ft(dt)1529 1516 y Fr(\024)1648 1448 y Fx(1)p 1645 1493 V 1645 1584 a Ft(h)1727 1380 y Fj(Z)1827 1406 y Fu(y)1783 1606 y(x)1885 1516 y Fx(2)1934 1474 y Fv(\000)p Fu(k)2048 1516 y Ft(dt)27 b Fx(=)h(2)2314 1474 y Fv(\000)p Fu(k)2411 1516 y Ft(:)38 1780 y Fx(This)34 b(w)m(as)h(to)f(b)s(e)h(sho)m(wn.)50 b(No)m(w)34 b(let)g Ft(G)g Fx(b)s(e)g(an)m(y)h(di\013eren)m(tiable)e (function)h(on)g([)p Ft(a;)17 b(b)p Fx(])35 b(with)f Ft(G)3477 1744 y Fv(0)3531 1780 y Fx(=)c Ft(f)11 b Fx(.)38 1896 y(Then)46 b(\()p Ft(F)f Fr(\000)31 b Ft(G)p Fx(\))674 1860 y Fv(0)747 1896 y Fx(=)49 b Ft(F)949 1860 y Fv(0)1003 1896 y Fr(\000)32 b Ft(G)1189 1860 y Fv(0)1262 1896 y Fx(=)49 b Ft(f)42 b Fr(\000)31 b Ft(f)60 b Fx(=)50 b(0.)82 b(Hence)46 b(b)m(y)h(the)f(Mean)g(V)-8 b(alue)44 b(Theorem)i(4.8)38 2013 y(\()p Ft(F)36 b Fr(\000)22 b Ft(G)p Fx(\)\()p Ft(c)p Fx(\))28 b(=)f(\()p Ft(F)36 b Fr(\000)23 b Ft(G)p Fx(\)\()p Ft(d)p Fx(\))32 b(for)g(an)m(y)h Ft(c;)17 b(d)32 b Fx(in)g([)p Ft(a;)17 b(b)p Fx(].)p 3682 1954 41 4 v 3682 2009 4 56 v 3720 2009 V 3682 2012 41 4 v 38 2241 a Fp(Theorem)39 b(5.4)i Fx(\(F)-8 b(undamen)m(tal)33 b(Theorem)i(of)f(Calculus\))p Fp(.)43 b Fw(L)-5 b(et)37 b Ft(f)22 b Fx(:)34 b Ft(I)39 b Fr(!)31 b Fn(R)48 b Fw(b)-5 b(e)37 b(c)-5 b(ontinuous)36 b(and)38 2357 y Ft(F)25 b Fx(:)33 b Ft(I)i Fr(!)28 b Fn(R)46 b Fw(such)34 b(that)h Ft(F)996 2321 y Fv(0)1047 2357 y Fx(=)28 b Ft(f)11 b Fw(.)44 b(Then)34 b(for)h(al)5 b(l)34 b Ft(a;)17 b(b)28 b Fr(2)g Ft(I)1310 2496 y Fj(Z)1409 2523 y Fu(b)1365 2722 y(a)1460 2632 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)28 b Fx(=)f Ft(F)14 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(F)14 b Fx(\()p Ft(a)p Fx(\))p Ft(:)38 2912 y Fw(Pr)-5 b(o)g(of.)41 b Fx(F)-8 b(or)32 b Ft(x)c Fr(2)g Ft(I)40 b Fx(de\014ne)1454 3128 y Ft(F)1517 3143 y Fm(0)1557 3128 y Fx(\()p Ft(x)p Fx(\))28 b(:=)1847 2993 y Fj(Z)1946 3019 y Fu(x)1902 3218 y(a)2007 3128 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt:)38 3371 y Fx(By)33 b(Theorem)g(5.3)f(w)m(e)i(ha)m(v)m(e)f Ft(F)1205 3335 y Fv(0)1191 3395 y Fm(0)1259 3371 y Fx(=)27 b Ft(f)11 b Fx(.)43 b(Clearly)1001 3651 y Ft(F)1064 3666 y Fm(0)1104 3651 y Fx(\()p Ft(a)p Fx(\))28 b(=)f(0)195 b(and)g Ft(F)2021 3666 y Fm(0)2061 3651 y Fx(\()p Ft(b)p Fx(\))28 b(=)2309 3515 y Fj(Z)2409 3541 y Fu(b)2365 3741 y(a)2460 3651 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt:)38 3914 y Fx(Hence)37 b(for)e(an)m(y)i Ft(F)24 b Fx(:)35 b Ft(I)41 b Fr(!)32 b Fn(R)47 b Fx(suc)m(h)37 b(that)f Ft(F)1659 3877 y Fv(0)1715 3914 y Fx(=)d Ft(f)11 b Fx(,)36 b(again)f(b)m(y)h(Theorem)g(5.3)g(there)g(is)f(a)h(constan)m(t)38 4030 y Ft(c)27 b Fr(2)i Fn(R)43 b Fx(suc)m(h)34 b(that)e Ft(F)k Fr(\000)23 b Ft(F)999 4045 y Fm(0)1066 4030 y Fx(=)28 b Ft(c)p Fx(.)43 b(Therefore)803 4300 y Ft(F)14 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(F)14 b Fx(\()p Ft(a)p Fx(\))27 b(=)h Ft(F)1517 4315 y Fm(0)1556 4300 y Fx(\()p Ft(b)p Fx(\))23 b Fr(\000)g Ft(F)1859 4315 y Fm(0)1898 4300 y Fx(\()p Ft(a)p Fx(\))28 b(=)f Ft(F)2219 4315 y Fm(0)2259 4300 y Fx(\()p Ft(b)p Fx(\))h(=)2508 4164 y Fj(Z)2607 4191 y Fu(b)2563 4390 y(a)2658 4300 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt;)38 4559 y Fx(whic)m(h)33 b(w)m(as)g(to)g(b)s(e)f(sho)m(wn.)p 3682 4500 V 3682 4556 4 56 v 3720 4556 V 3682 4559 41 4 v 137 4788 a(It)h(is)f(common)f(to)i(use)g(the)g(notation)997 5022 y Ft(F)14 b Fx(\()p Ft(x)p Fx(\))1205 4907 y Fj(\014)1205 4967 y(\014)1205 5027 y(\014)1238 4934 y Fu(b)1238 5091 y(a)1377 5022 y Fx(or)1562 4911 y Fj(h)1609 5022 y Ft(F)g Fx(\()p Ft(x)p Fx(\))1817 4911 y Fj(i)1864 4934 y Fu(b)1864 5091 y(a)2003 5022 y Fx(for)97 b Ft(F)14 b Fx(\()p Ft(b)p Fx(\))22 b Fr(\000)h Ft(F)14 b Fx(\()p Ft(a)p Fx(\))p Ft(:)38 5260 y Fx(The)33 b(form)m(ula)e(from)g(Theorem)i(5.4)f(can)h (then)g(w)m(e)h(written)e(as)823 5380 y Fj(Z)923 5406 y Fu(b)879 5605 y(a)974 5515 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)27 b Fx(=)h Ft(F)14 b Fx(\()p Ft(x)p Fx(\))1626 5401 y Fj(\014)1626 5460 y(\014)1626 5520 y(\014)1659 5427 y Fu(b)1659 5584 y(a)1798 5515 y Fx(or)1999 5380 y Fj(Z)2099 5406 y Fu(b)2054 5605 y(a)2150 5515 y Ft(f)d Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)27 b Fx(=)2594 5405 y Fj(h)2641 5515 y Ft(F)14 b Fx(\()p Ft(x)p Fx(\))2849 5405 y Fj(i)2896 5427 y Fu(b)2896 5584 y(a)38 5774 y Fx(whic)m(h)33 b(is)f(often)h(abbreviated)f(to)1483 5833 y Fj(Z)1599 5969 y Ft(f)11 b Fx(\()p Ft(x)p Fx(\))17 b Ft(dx)27 b Fx(=)h Ft(F)14 b Fx(\()p Ft(x)p Fx(\))p Ft(:)38 6204 y Fx(Ho)m(w)m(ev)m(er,)29 b(this)c(notation)f(is)h (somewhat)g(problematic,)g(since)h Ft(F)39 b Fx(is)25 b(determined)g(up)h(to)f(a)g(constan)m(t)38 6320 y(only)-8 b(.)p eop %%Page: 18 18 18 17 bop 1208 450 a Fx(6.)48 b Fs(Sequences)37 b(of)h(Functions)137 624 y Fx(W)-8 b(e)39 b(de\014ne)g(the)f(notion)f(of)g(uniform)g(con)m (v)m(ergence)j(of)e(a)f(sequence)k(of)c(con)m(tin)m(uous)i(functions)38 741 y Ft(f)86 756 y Fu(i)125 741 y Fx(:)33 b Ft(I)j Fr(!)27 b Fn(R)44 b Fx(to)32 b(a)g(con)m(tin)m(uous)i(function)e Ft(f)21 b Fx(:)34 b Ft(I)h Fr(!)28 b Fn(R)t Fx(.)38 882 y Fp(De\014nition)41 b(6.1.)j Fx(Let)37 b Ft(f)1011 897 y Fu(i)1040 882 y Ft(;)17 b(f)k Fx(:)35 b Ft(I)43 b Fr(!)35 b Fn(R)48 b Fx(b)s(e)37 b(con)m(tin)m(uous,)i(with)e(appro)m(ximating)e (maps)h Ft(h)3400 897 y Fu(i)3429 882 y Ft(;)17 b(h)37 b Fx(and)38 998 y Fs(Ca)n(uchy)e Fx(mo)s(duli)30 b Ft(\013)842 1013 y Fu(i)870 998 y Ft(;)17 b(\013)q Fx(.)43 b(The)33 b(sequence)i(\()p Ft(f)1737 1013 y Fu(i)1766 998 y Fx(\))1804 1013 y Fu(i)p Fv(2)p Fo(N)1959 998 y Fx(is)d Fw(uniformly)j(c)-5 b(onver)g(gent)41 b Fx(to)32 b Ft(f)44 b Fx(if)735 1169 y Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(i)p Fr(\025)r Ft(p)p Fr(8)p Ft(a)1298 1084 y Fj(\014)1298 1144 y(\014)1331 1169 y Ft(h)1387 1184 y Fu(i)1416 1169 y Fx(\()p Ft(a;)17 b(\013)1611 1184 y Fu(i)1639 1169 y Fx(\()p Ft(k)25 b Fx(+)d(2\)\))g Fr(\000)g Ft(h)p Fx(\()p Ft(a;)17 b(\013)q Fx(\()p Ft(k)25 b Fx(+)d(2\)\))2686 1084 y Fj(\014)2686 1144 y(\014)2747 1169 y Fr(\024)28 b Fx(2)2901 1128 y Fv(\000)p Fu(k)2998 1169 y Ft(:)38 1335 y Fp(Lemma)k(6.2.)38 b Fw(L)-5 b(et)32 b Ft(f)855 1350 y Fu(i)883 1335 y Ft(;)17 b(f)k Fx(:)34 b Ft(I)h Fr(!)27 b Fn(R)42 b Fw(b)-5 b(e)31 b(c)-5 b(ontinuous,)31 b(with)g(appr)-5 b(oximating)29 b(maps)h Ft(h)3144 1350 y Fu(i)3173 1335 y Ft(;)17 b(h)p Fw(.)43 b(Then)30 b(the)38 1451 y(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent.)38 1593 y Fx(\(a\))41 b Fw(The)34 b(se)-5 b(quenc)g(e)34 b Fx(\()p Ft(f)885 1608 y Fu(i)913 1593 y Fx(\))951 1608 y Fu(i)p Fv(2)p Fo(N)1109 1593 y Fw(is)h(uniformly)g(c)-5 b(onver)g(gent)34 b(to)h Ft(f)11 b Fw(.)32 1709 y Fx(\(b\))42 b Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(i;)17 b(n)p Fr(\025)r Ft(p)p Fr(8)p Ft(a)28 b Fr(j)p Ft(h)953 1724 y Fu(i)981 1709 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)p Fx(\()p Ft(a;)17 b(n)p Fx(\))p Fr(j)27 b(\024)h Fx(2)1826 1673 y Fv(\000)p Fu(k)1924 1709 y Fw(.)38 1887 y(Pr)-5 b(o)g(of.)41 b Fx(\(a\))32 b Fr(\))g Fx(\(b\).)650 1970 y Fj(\014)650 2030 y(\014)683 2055 y Ft(h)739 2070 y Fu(i)768 2055 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)p Fx(\()p Ft(a;)17 b(n)p Fx(\))1403 1970 y Fj(\014)1403 2030 y(\014)1464 2055 y Fr(\024)1570 1970 y Fj(\014)1570 2030 y(\014)1603 2055 y Ft(h)1659 2070 y Fu(i)1687 2055 y Fx(\()p Ft(a;)g(n)p Fx(\))22 b Fr(\000)h Ft(h)2094 2070 y Fu(i)2122 2055 y Fx(\()p Ft(a;)17 b(\013)2317 2070 y Fu(i)2345 2055 y Fx(\()p Ft(k)25 b Fx(+)d(3\)\))2682 1970 y Fj(\014)2682 2030 y(\014)2737 2055 y Fx(+)1570 2131 y Fj(\014)1570 2191 y(\014)1603 2216 y Ft(h)1659 2231 y Fu(i)1687 2216 y Fx(\()p Ft(a;)17 b(\013)1882 2231 y Fu(i)1910 2216 y Fx(\()p Ft(k)25 b Fx(+)d(3\)\))g Fr(\000)h Ft(h)p Fx(\()p Ft(a;)17 b(\013)q Fx(\()p Ft(k)25 b Fx(+)d(3\)\))2958 2131 y Fj(\014)2958 2191 y(\014)3013 2216 y Fx(+)1570 2292 y Fj(\014)1570 2352 y(\014)1603 2377 y Ft(h)p Fx(\()p Ft(a;)17 b(\013)q Fx(\()p Ft(k)25 b Fx(+)d(3\)\))f Fr(\000)i Ft(h)p Fx(\()p Ft(a;)17 b(n)p Fx(\))2598 2292 y Fj(\014)2598 2352 y(\014)1464 2540 y Fr(\024)29 b Fx(2)1619 2499 y Fv(\000)p Fu(k)r Fv(\000)p Fm(3)1828 2540 y Fx(+)22 b(2)1975 2499 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2185 2540 y Fx(+)g(2)2332 2499 y Fv(\000)p Fu(k)r Fv(\000)p Fm(3)38 2712 y Fx(if)36 b Ft(n)h Fr(\025)g Ft(\013)403 2727 y Fu(i)432 2712 y Fx(\()p Ft(k)28 b Fx(+)e(3\))p Ft(;)17 b(\013)q Fx(\()p Ft(k)28 b Fx(+)d(3\))38 b(\(for)f(the)h(\014rst)g(and)g(last)f(term\))g (and)h Ft(i)f Fr(\025)g Ft(p)h Fx(with)f Ft(p)h Fx(pro)m(vided)g(for)38 2829 y Ft(k)25 b Fx(+)d(1)32 b(b)m(y)j(\(a\).)137 2945 y(\(b\))e Fr(\))f Fx(\(a\).)471 3028 y Fj(\014)471 3088 y(\014)504 3113 y Ft(h)560 3128 y Fu(i)588 3113 y Fx(\()p Ft(a;)17 b(\013)783 3128 y Fu(i)811 3113 y Fx(\()p Ft(k)25 b Fx(+)d(2\)\))g Fr(\000)h Ft(h)p Fx(\()p Ft(a;)17 b(\013)q Fx(\()p Ft(k)25 b Fx(+)d(2\)\))1859 3028 y Fj(\014)1859 3088 y(\014)1919 3113 y Fr(\024)2024 3028 y Fj(\014)2024 3088 y(\014)2057 3113 y Ft(h)2113 3128 y Fu(i)2142 3113 y Fx(\()p Ft(a;)17 b(\013)2337 3128 y Fu(i)2365 3113 y Fx(\()p Ft(k)25 b Fx(+)d(2\)\))g Fr(\000)g Ft(h)2879 3128 y Fu(i)2908 3113 y Fx(\()p Ft(a;)17 b(n)p Fx(\))3137 3028 y Fj(\014)3137 3088 y(\014)3192 3113 y Fx(+)2024 3189 y Fj(\014)2024 3249 y(\014)2057 3274 y Ft(h)2113 3289 y Fu(i)2142 3274 y Fx(\()p Ft(a;)g(n)p Fx(\))22 b Fr(\000)h Ft(h)p Fx(\()p Ft(a;)17 b(n)p Fx(\))2778 3189 y Fj(\014)2778 3249 y(\014)2833 3274 y Fx(+)2024 3350 y Fj(\014)2024 3410 y(\014)2057 3435 y Ft(h)p Fx(\()p Ft(a;)g(n)p Fx(\))23 b Fr(\000)f Ft(h)p Fx(\()p Ft(a;)17 b(\013)q Fx(\()p Ft(k)25 b Fx(+)d(2\)\))3053 3350 y Fj(\014)3053 3410 y(\014)1919 3599 y Fr(\024)28 b Fx(2)2073 3558 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)2283 3599 y Fx(+)22 b(2)2430 3558 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)2640 3599 y Fx(+)g(2)2787 3558 y Fv(\000)p Fu(k)r Fv(\000)p Fm(2)38 3771 y Fx(if)34 b Ft(n)f Fr(\025)g Ft(\013)393 3786 y Fu(i)422 3771 y Fx(\()p Ft(k)27 b Fx(+)d(2\))p Ft(;)17 b(\013)q Fx(\()p Ft(k)26 b Fx(+)e(2\))35 b(\(for)g(the)h(\014rst)g(and)g(last)f(term\))g (and)g(in)g(addition)f Ft(i;)17 b(n)33 b Fr(\025)g Ft(p)j Fx(with)f Ft(p)38 3887 y Fx(pro)m(vided)e(for)f Ft(k)25 b Fx(+)d(1)32 b(b)m(y)j(\(b\).)p 3682 3828 41 4 v 3682 3884 4 56 v 3720 3884 V 3682 3887 41 4 v 137 4065 a(The)24 b(notion)e(of)h(uniform)e(con)m(v)m(ergence)k(has)f(b)s(een)f (de\014ned)i(in)d(terms)h(of)f(witnesses)j(for)d(the)i(giv)m(en)38 4181 y(con)m(tin)m(uous)40 b(functions)f Ft(f)1008 4196 y Fu(i)1036 4181 y Fx(,)i(in)d(order)i(to)e(ensure)j(that)e(from)f(a)h (pro)s(of)f(of)h(uniform)e(con)m(v)m(ergence)38 4297 y(w)m(e)42 b(can)f(extract)g(the)h(righ)m(t)e(data.)68 b(The)42 b(next)g(Lemma)d(sho)m(ws)k(that)d(a)h(uniformly)e(con)m(v)m (ergen)m(t)38 4413 y(sequence)c(indeed)e(is)f(uniformly)e(con)m(v)m (ergen)m(t)35 b(in)d(the)h(usual)f(sense.)38 4555 y Fp(Lemma)51 b(6.3.)d Fw(L)-5 b(et)47 b Ft(f)899 4570 y Fu(i)927 4555 y Ft(;)17 b(f)k Fx(:)38 b Ft(I)56 b Fr(!)48 b Fn(R)57 b Fw(b)-5 b(e)46 b(c)-5 b(ontinuous.)79 b(Assume)46 b(that)g(the)g(se) -5 b(quenc)g(e)46 b Fx(\()p Ft(f)3418 4570 y Fu(i)3446 4555 y Fx(\))3484 4570 y Fu(i)p Fv(2)p Fo(N)3653 4555 y Fw(is)38 4671 y(uniformly)34 b(c)-5 b(onver)g(gent)34 b(to)h Ft(f)11 b Fw(.)45 b(Then)1151 4837 y Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(i)p Fr(\025)r Ft(p)p Fr(8)p Ft(x)1718 4753 y Fj(\014)1718 4812 y(\014)1751 4837 y Ft(f)1799 4852 y Fu(i)1827 4837 y Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)f Ft(f)11 b Fx(\()p Ft(x)p Fx(\))2270 4753 y Fj(\014)2270 4812 y(\014)2331 4837 y Fr(\024)29 b Fx(2)2486 4796 y Fv(\000)p Fu(k)2583 4837 y Ft(:)38 5017 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)35 b Ft(x)d Fx(=)f(\(\()p Ft(a)834 5032 y Fu(n)881 5017 y Fx(\))919 5032 y Fu(n)966 5017 y Ft(;)17 b(M)10 b Fx(\),)35 b(and)g(let)f Ft(h)1605 5032 y Fu(i)1633 5017 y Ft(;)17 b(h)35 b Fx(b)s(e)g(appro)m(ximating)d(maps)j(for)f Ft(f)3012 5032 y Fu(i)3040 5017 y Ft(;)17 b(f)11 b Fx(,)35 b(resp)s(ectiv)m(ely)-8 b(.)38 5133 y(By)33 b(Lemma)e(6.2\(b\))1042 5263 y Fr(8)p Ft(k)s Fr(9)p Ft(p)p Fr(8)p Ft(n)p Fr(\025)r Ft(p)d Fr(j)p Ft(h)1608 5278 y Fu(n)1655 5263 y Fx(\()p Ft(a)1744 5278 y Fu(n)1791 5263 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)p Fx(\()p Ft(a)2197 5278 y Fu(n)2245 5263 y Ft(;)17 b(n)p Fx(\))p Fr(j)27 b(\024)h Fx(2)2594 5222 y Fv(\000)p Fu(k)2691 5263 y Ft(;)38 5409 y Fx(whence)34 b(the)f(claim.)p 3682 5350 V 3682 5405 4 56 v 3720 5405 V 3682 5408 41 4 v 137 5586 a(The)38 b(next)f(Lemma)d(giv)m(es)j(a)f (useful)g(criterion)f(as)h(to)g(when)h(and)g(ho)m(w)f(w)m(e)i(can)e (construct)h(the)38 5703 y(limit)29 b(function.)43 b(It)32 b(will)f(b)s(e)i(used)g(in)f(the)h(pro)s(of)f(of)g(Theorem)h(7.7.)38 5844 y Fp(Lemma)h(6.4.)39 b Fw(L)-5 b(et)33 b Ft(f)859 5859 y Fu(i)898 5844 y Fx(:)h Ft(I)h Fr(!)27 b Fn(R)44 b Fw(b)-5 b(e)31 b(c)-5 b(ontinuous)33 b(functions,)f(given)f(by)i (appr)-5 b(oximating)30 b(functions)38 5960 y Ft(h)94 5975 y Fu(i)160 5960 y Fw(and)38 b(mo)-5 b(duli)37 b Ft(\013)i Fw(of)59 b Fs(Ca)n(uchy)p Fw(ness)38 b(and)f Ft(!)42 b Fw(of)37 b(\(uniform\))h(c)-5 b(ontinuity,)39 b(wher)-5 b(e)37 b(the)h(latter)h(two)38 6076 y(ar)-5 b(e)46 b Fx(indep)s(enden)m(t)41 b Fw(of)e Ft(i)p Fw(.)57 b(Assume)40 b(we)e(have)h(a)g(mo)-5 b(dulus)39 b Ft(\016)15 b Fx(:)35 b Fn(N)50 b Fr(!)36 b Fn(N)54 b Fw(of)60 b Fx(uniform)35 b(con)m(v)m(ergence)38 6193 y Fw(satisfying)1329 6233 y Fj(\014)1329 6293 y(\014)1362 6318 y Ft(h)1418 6333 y Fu(i)1447 6318 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)g Ft(h)1853 6333 y Fu(j)1890 6318 y Fx(\()p Ft(a;)17 b(n)p Fx(\))2119 6233 y Fj(\014)2119 6293 y(\014)2180 6318 y Fr(\024)28 b Fx(2)2334 6277 y Fv(\000)p Fu(k)p eop %%Page: 19 19 19 18 bop 38 450 a Fw(for)39 b Ft(i;)17 b(j)41 b Fr(\025)36 b Ft(\016)t Fx(\()p Ft(k)s Fx(\))k Fw(and)e Ft(n)e Fr(\025)g Ft(\013)q Fx(\()p Ft(k)s Fx(\))p Fw(,)k(and)f(al)5 b(l)38 b Ft(a)e Fr(2)g Ft(I)8 b Fw(.)58 b(Then)38 b Fx(\()p Ft(f)2358 465 y Fu(i)2386 450 y Fx(\))2424 465 y Fu(i)p Fv(2)p Fo(N)2587 450 y Fw(uniformly)h(c)-5 b(onver)g(ges)37 b(to)j(the)38 566 y(c)-5 b(ontinuous)34 b(function)h Ft(f)22 b Fx(:)33 b Ft(I)j Fr(!)27 b Fn(R)46 b Fw(given)34 b(by)439 723 y Ft(h)495 738 y Fu(f)540 723 y Fx(\()p Ft(a;)17 b(n)p Fx(\))28 b(:=)g Ft(h)984 738 y Fu(n)1031 723 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Ft(;)116 b(\013)1465 738 y Fu(f)1510 723 y Fx(\()p Ft(k)s Fx(\))28 b(:=)g(max)1980 643 y Fj(\000)2026 723 y Ft(\016)t Fx(\()p Ft(k)d Fx(+)d(1\))p Ft(;)17 b(\013)q Fx(\()p Ft(k)24 b Fx(+)e(1\))2777 643 y Fj(\001)2823 723 y Ft(;)116 b(!)3027 738 y Fu(f)3100 723 y Fx(:=)27 b Ft(!)t(:)38 889 y Fw(Pr)-5 b(o)g(of.)41 b Fx(It)36 b(is)g(easy)h(to)f(see)i(that)e(this)g(function)f Ft(f)22 b Fx(:)35 b Ft(I)41 b Fr(!)34 b Fn(R)47 b Fx(giv)m(en)36 b(b)m(y)i Ft(h)2756 904 y Fu(f)2801 889 y Fx(,)f Ft(\013)2927 904 y Fu(f)3009 889 y Fx(and)f Ft(!)3263 904 y Fu(f)3345 889 y Fx(is)f(indeed)38 1006 y(con)m(tin)m(uous:)44 b Ft(\013)626 1021 y Fu(f)704 1006 y Fx(is)32 b(a)g Fs(Ca)n(uchy)j Fx(mo)s(dulus,)d(b)s(ecause)425 1157 y Fr(j)p Ft(h)509 1172 y Fu(n)556 1157 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)963 1172 y Fu(m)1030 1157 y Fx(\()p Ft(a;)17 b(m)p Fx(\))p Fr(j)27 b(\024)h(j)p Ft(h)1530 1172 y Fu(n)1577 1157 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)1984 1172 y Fu(m)2051 1157 y Fx(\()p Ft(a;)17 b(n)p Fx(\))p Fr(j)k Fx(+)h Fr(j)p Ft(h)2511 1172 y Fu(m)2578 1157 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)2985 1172 y Fu(m)3051 1157 y Fx(\()p Ft(a;)17 b(m)p Fx(\))p Fr(j)1341 1321 y(\024)28 b Fx(2)1495 1280 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1705 1321 y Fx(+)22 b(2)1852 1280 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)38 1483 y Fx(for)32 b Ft(m;)17 b(n)28 b Fr(\025)g Ft(\013)569 1498 y Fu(f)614 1483 y Fx(\()p Ft(k)s Fx(\),)33 b(and)g Ft(!)i Fx(is)e(a)f(mo)s(dulus)f(of)h(\(uniform\))f(con)m(tin)m(uit)m(y) -8 b(,)33 b(b)s(ecause)877 1651 y Fr(j)p Ft(a)22 b Fr(\000)h Ft(b)p Fr(j)28 b(\024)g Fx(2)1329 1610 y Fv(\000)p Fu(!)r Fm(\()p Fu(k)r Fm(\)+1)1645 1651 y Fr(!)g(j)p Ft(h)1857 1666 y Fu(n)1903 1651 y Fx(\()p Ft(a;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)2310 1666 y Fu(n)2357 1651 y Fx(\()p Ft(b;)17 b(n)p Fx(\))p Fr(j)28 b(\024)g Fx(2)2786 1610 y Fv(\000)p Fu(k)38 1808 y Fx(for)37 b Ft(n)h Fr(\025)g Ft(\013)465 1823 y Fu(f)510 1808 y Fx(\()p Ft(k)s Fx(\).)60 b(Moreo)m(v)m(er,)42 b(for)37 b(a)h(giv)m(en)h Ft(k)i Fx(w)m(e)e(ma)m(y)f(pic)m(k)h Ft(p)e Fx(:=)g(max\()p Ft(\016)t Fx(\()p Ft(k)s Fx(\))p Ft(;)17 b(\013)q Fx(\()p Ft(k)s Fx(\)\).)59 b(Then)39 b(for)38 1927 y Ft(i;)17 b(n)28 b Fr(\025)g Ft(p)i Fx(and)g(all)e Ft(a)p Fx(,)i Fr(j)p Ft(h)897 1942 y Fu(i)926 1927 y Fx(\()p Ft(a;)17 b(n)p Fx(\))f Fr(\000)i Ft(h)1322 1942 y Fu(n)1369 1927 y Fx(\()p Ft(a;)f(n)p Fx(\))p Fr(j)27 b(\024)h Fx(2)1807 1890 y Fv(\000)p Fu(k)1905 1927 y Fx(.)42 b(By)31 b(Lemma)d(6.2\(b\))i (this)g(means)g(that)g(\()p Ft(f)3534 1942 y Fu(i)3562 1927 y Fx(\))3600 1942 y Fu(i)p Fv(2)p Fo(N)38 2043 y Fx(uniformly)g(con)m(v)m(erges)35 b(to)d Ft(f)11 b Fx(.)p 3682 1984 41 4 v 3682 2040 4 56 v 3720 2040 V 3682 2043 41 4 v 137 2209 a(W)-8 b(e)36 b(no)m(w)g(sho)m(w)h(that)e(for)f(a)h (uniformly)f(con)m(v)m(ergen)m(t)j(sequence)h(of)d(con)m(tin)m(uous)h (functions,)g(in-)38 2325 y(tegration)31 b(and)i(limits)c(can)k(b)s(e)g (exc)m(hanged.)38 2462 y Fp(Theorem)h(6.5.)39 b Fw(L)-5 b(et)33 b Ft(f)937 2477 y Fu(i)965 2462 y Ft(;)17 b(f)22 b Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)f Fn(R)44 b Fw(b)-5 b(e)31 b(c)-5 b(ontinuous.)44 b(Assume)32 b(that)h(the)f(se) -5 b(quenc)g(e)32 b Fx(\()p Ft(f)3432 2477 y Fu(i)3460 2462 y Fx(\))3498 2477 y Fu(i)p Fv(2)p Fo(N)3653 2462 y Fw(is)38 2578 y(uniformly)i(c)-5 b(onver)g(gent)34 b(to)h Ft(f)11 b Fw(.)45 b(Then)1293 2802 y Fx(lim)1279 2862 y Fu(i)p Fv(!1)1460 2666 y Fj(Z)1560 2692 y Fu(b)1516 2892 y(a)1611 2802 y Ft(f)1659 2817 y Fu(i)1687 2802 y Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)27 b Fx(=)2032 2666 y Fj(Z)2132 2692 y Fu(b)2087 2892 y(a)2183 2802 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt:)38 3019 y Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)280 3107 y Fj(Z)380 3133 y Fu(b)336 3333 y(a)431 3243 y Ft(f)479 3258 y Fu(i)507 3243 y Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)27 b Fx(=)h(\()p Ft(S)6 b Fx(\()p Ft(f)1042 3258 y Fu(i)1070 3243 y Ft(;)17 b(a;)g(b;)g(n)p Fx(\)\))1428 3258 y Fu(n)1502 3243 y Fx(=:)28 b Ft(x)1688 3258 y Fu(i)1814 3243 y Fx(and)2086 3107 y Fj(Z)2185 3133 y Fu(b)2141 3333 y(a)2236 3243 y Ft(f)11 b Fx(\()p Ft(t)p Fx(\))17 b Ft(dt)27 b Fx(=)h(\()p Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\)\))3193 3258 y Fu(n)3267 3243 y Fx(=:)28 b Ft(x:)38 3456 y Fx(W)-8 b(e)33 b(sho)m(w)g(that)g(\() p Ft(x)752 3471 y Fu(i)780 3456 y Fx(\))818 3471 y Fu(i)879 3456 y Fx(con)m(v)m(erges)i(to)d Ft(x)p Fx(,)h(using)f(Lemma)g (2.4\(b\))o(.)604 3623 y Fj(\014)604 3683 y(\014)637 3708 y Ft(S)6 b Fx(\()p Ft(f)789 3723 y Fu(i)817 3708 y Ft(;)17 b(a;)g(b;)g(n)p Fx(\))22 b Fr(\000)h Ft(S)6 b Fx(\()p Ft(f)f(;)17 b(a;)g(b;)g(n)p Fx(\))1736 3623 y Fj(\014)1736 3683 y(\014)1797 3708 y Fr(\024)1912 3640 y Ft(b)23 b Fr(\000)f Ft(a)p 1912 3685 215 4 v 1990 3776 a(n)2158 3583 y Fu(n)p Fv(\000)p Fm(1)2153 3613 y Fj(X)2164 3823 y Fu(j)t Fm(=0)2297 3623 y Fj(\014)2297 3683 y(\014)2330 3708 y Ft(h)2386 3723 y Fu(i)2414 3708 y Fx(\()p Ft(a)2503 3723 y Fu(j)2540 3708 y Ft(;)17 b(n)p Fx(\))22 b Fr(\000)h Ft(h)p Fx(\()p Ft(a)2947 3723 y Fu(j)2984 3708 y Ft(;)17 b(n)p Fx(\))3124 3623 y Fj(\014)3124 3683 y(\014)1797 4020 y Fr(\024)1912 3952 y Ft(b)23 b Fr(\000)f Ft(a)p 1912 3997 V 1990 4088 a(n)2136 4020 y(n)p Fx(2)2243 3979 y Fv(\000)p Fu(l)2352 4020 y Fr(\024)28 b Fx(2)2506 3979 y Fv(\000)p Fu(k)2603 4020 y Ft(;)38 4225 y Fx(for)h Ft(n;)17 b(i)28 b Fr(\025)g Ft(p)501 4240 y Fu(l)527 4225 y Fx(,)j(where)g Ft(a)915 4240 y Fu(j)979 4225 y Fx(:=)d Ft(a)17 b Fx(+)1285 4181 y Fu(j)p 1280 4202 43 4 v 1280 4259 a(n)1333 4225 y Fx(\()p Ft(b)g Fr(\000)g Ft(a)p Fx(\).)42 b Ft(p)1730 4240 y Fu(l)1786 4225 y Fx(is)29 b(de\014ned)j(in)d(Lemma)f(6.2\(b\),)i(and)g Ft(l)i Fx(is)e(suc)m(h)h(that)38 4350 y(\()p Ft(b)22 b Fr(\000)h Ft(a)p Fx(\)2)377 4314 y Fv(\000)p Fu(l)485 4350 y Fr(\024)29 b Fx(2)640 4314 y Fv(\000)p Fu(k)737 4350 y Fx(.)p 3682 4291 41 4 v 3682 4346 4 56 v 3720 4346 V 3682 4349 41 4 v 961 4547 a(7.)49 b Fs(Ordinar)-7 b(y)39 b(Differential)e(Equa)-7 b(tions)137 4722 y Fx(Consider)33 b(a)g(di\013eren)m(tial)d(equation)38 4879 y(\(10\))1410 b Ft(y)1674 4837 y Fv(0)1724 4879 y Fx(=)27 b Ft(f)11 b Fx(\()p Ft(x;)17 b(y)t Fx(\))p Ft(;)38 5035 y Fx(where)35 b Ft(f)22 b Fx(:)33 b Ft(D)g Fr(!)c Fn(R)45 b Fx(is)33 b(a)h(con)m(tin)m(uous)g(function)g(on)f(some)h(subset)i Ft(D)g Fx(of)d Fn(R)2833 4999 y Fm(2)2879 5035 y Fx(.)47 b(A)34 b(solution)e(of)40 b(\(10\))38 5152 y(on)c(an)g(in)m(terv)-5 b(al)34 b Ft(I)44 b Fx(is)36 b(a)g(function)f Ft(')11 b Fx(:)35 b Ft(I)41 b Fr(!)33 b Fn(R)47 b Fx(with)36 b(a)g(con)m(tin)m(uous)g(deriv)-5 b(ativ)m(e)36 b Ft(')3109 5116 y Fv(0)3168 5152 y Fx(suc)m(h)h(that)f(for)38 5268 y(all)30 b Ft(x)e Fr(2)g Ft(I)457 5425 y Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))28 b Fr(2)g Ft(D)36 b Fx(\(hence)d Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))33 b(is)f(de\014ned\))i(and)f Ft(')2562 5389 y Fv(0)2585 5425 y Fx(\()p Ft(x)p Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\).)137 5582 y(W)-8 b(e)34 b(w)m(an)m(t)h(to)e(construct)i(appro)m(ximate)d(solutions)h(to) g(\(10\))o(,)h(b)m(y)g(means)g(of)f(the)h(the)g Fs(Ca)n(uchy)r Fx(-)38 5698 y Fs(Euler)e Fx(appro)m(ximation)e(metho)s(d.)38 5835 y Fp(De\014nition)37 b(7.1.)42 b Fx(Let)34 b Ft(f)22 b Fx(:)33 b Ft(D)f Fr(!)d Fn(R)44 b Fx(b)s(e)34 b(con)m(tin)m(uous,)h (and)e(consider)h(an)g(in)m(terv)-5 b(al)32 b Ft(I)8 b Fx(.)46 b(A)33 b(function)38 5951 y Ft(')11 b Fx(:)33 b Ft(I)j Fr(!)27 b Fn(R)43 b Fx(is)32 b(a)h Fw(solution)h(up)h(to)g Fx(2)1334 5915 y Fv(\000)p Fu(k)1464 5951 y Fx(of)k(\(10\))32 b(if)38 6087 y(\(a\))41 b Ft(')32 b Fx(is)g Fw(admissible)7 b Fx(,)32 b(i.e.,)g(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))28 b Fr(2)g Ft(D)35 b Fx(for)d Ft(x)d Fr(2)f Ft(I)8 b Fx(.)32 6204 y(\(b\))42 b Ft(')32 b Fx(is)g(con)m(tin)m(uous.) 43 6320 y(\(c\))42 b Ft(')32 b Fx(has)h(a)g(piecewise)g(con)m(tin)m (uous)g(deriv)-5 b(ativ)m(e)32 b(on)h Ft(I)8 b Fx(.)p eop %%Page: 20 20 20 19 bop 32 450 a Fx(\(d\))42 b Fr(j)p Ft(')296 414 y Fv(0)319 450 y Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))p Fr(j)27 b(\024)h Fx(2)1210 414 y Fv(\000)p Fu(k)1340 450 y Fx(for)k(all)f Ft(x)d Fr(2)g Ft(I)40 b Fx(where)34 b Ft(')2231 414 y Fv(0)2254 450 y Fx(\()p Ft(x)p Fx(\))f(is)f (de\014ned.)38 594 y Fp(Theorem)39 b(7.2.)k Fw(L)-5 b(et)37 b Ft(f)22 b Fx(:)34 b Ft(D)g Fr(!)c Fn(R)48 b Fw(b)-5 b(e)36 b(c)-5 b(ontinuous,)37 b(and)f Fx(\()p Ft(a)2311 609 y Fm(0)2351 594 y Ft(;)17 b(b)2436 609 y Fm(0)2475 594 y Fx(\))31 b Fr(2)h Ft(D)39 b Fw(such)e(that)g(the)f(r)-5 b(e)g(ctangle)38 710 y Ft(R)33 b Fw(given)e(by)i Fr(j)p Ft(x)16 b Fr(\000)g Ft(a)764 725 y Fm(0)804 710 y Fr(j)28 b(\024)g Ft(a)p Fw(,)33 b Fr(j)p Ft(y)19 b Fr(\000)d Ft(b)1308 725 y Fm(0)1348 710 y Fr(j)28 b(\024)g Ft(b)k Fw(is)g(in)g Ft(D)s Fw(.)44 b(Assume)32 b Fr(j)p Ft(f)11 b Fx(\()p Ft(x;)17 b(y)t Fx(\))p Fr(j)26 b(\024)i Ft(M)43 b Fw(for)32 b Fx(\()p Ft(x;)17 b(y)t Fx(\))27 b Fr(2)h Ft(R)q Fw(,)k(and)38 827 y(let)k Ft(h)29 b Fx(:=)g(min)n(\()p Ft(a;)17 b(b=)-5 b(M)10 b Fx(\))p Fw(.)48 b(Then)34 b(for)i(every)f Ft(k)d Fr(2)e Fn(N)50 b Fw(we)36 b(c)-5 b(an)35 b(c)-5 b(onstruct)35 b(an)h(appr)-5 b(oximate)34 b(solution)38 943 y Ft(')11 b Fx(:)33 b([)p Ft(a)251 958 y Fm(0)313 943 y Fr(\000)23 b Ft(h;)17 b(a)564 958 y Fm(0)625 943 y Fx(+)22 b Ft(h)p Fx(])28 b Fr(!)f Fn(R)46 b Fw(of)56 b Fx(\(10\))34 b Fw(up)h(to)g Fx(2)1715 907 y Fv(\000)p Fu(k)1847 943 y Fw(such)g(that)g Ft(')p Fx(\()p Ft(a)2421 958 y Fm(0)2461 943 y Fx(\))27 b(=)h Ft(b)2671 958 y Fm(0)2711 943 y Fw(.)38 1127 y(Pr)-5 b(o)g(of.)41 b Fx(By)33 b(de\014nition)f(of)g Ft(h)p Fx(,)g(the)h(rectangle)1049 1298 y Ft(S)38 b Fx(giv)m(en)33 b(b)m(y)g Fr(j)p Ft(x)22 b Fr(\000)h Ft(a)1793 1313 y Fm(0)1833 1298 y Fr(j)k(\024)h Ft(h)p Fx(,)33 b Fr(j)p Ft(y)25 b Fr(\000)e Ft(b)2351 1313 y Fm(0)2390 1298 y Fr(j)28 b(\024)g Ft(M)10 b(h)38 1470 y Fx(is)36 b(in)h Ft(D)s Fx(.)57 b Ft(f)48 b Fx(is)36 b(con)m(tin)m(uous,)k(hence)e(comes)g(with)e(a)h(mo)s(dulus)f(of)h (\(uniform\))e(con)m(tin)m(uit)m(y;)40 b(so)d(for)38 1586 y(our)32 b(giv)m(en)h Ft(k)j Fx(w)m(e)d(ha)m(v)m(e)h(an)f Ft(l)h Fx(suc)m(h)g(that)38 1758 y(\(11\))1110 b Fr(j)p Ft(f)11 b Fx(\()6 b(~)-55 b Ft(x;)23 b Fx(~)-55 b Ft(y)s Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b(y)t Fx(\))p Fr(j)26 b(\024)i Fx(2)2251 1717 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)38 1930 y Fx(for)k(\()6 b(~)-55 b Ft(x;)24 b Fx(~)-56 b Ft(y)s Fx(\))p Ft(;)17 b Fx(\()p Ft(x;)g(y)t Fx(\))31 b(in)h Ft(D)j Fx(and)e Fr(j)6 b Fx(~)-55 b Ft(x)22 b Fr(\000)h Ft(x)p Fr(j)p Ft(;)17 b Fr(j)7 b Fx(~)-56 b Ft(y)25 b Fr(\000)d Ft(y)t Fr(j)27 b(\024)h Fx(2)1928 1894 y Fv(\000)p Fu(l)2009 1930 y Fx(.)137 2046 y(W)-8 b(e)33 b(no)m(w)h(divide)e(the)h(in)m(terv)-5 b(al)31 b([)p Ft(a)1397 2061 y Fm(0)1437 2046 y Ft(;)17 b(a)1532 2061 y Fm(0)1593 2046 y Fx(+)22 b Ft(h)p Fx(])33 b(suc)m(h)h(that)38 2190 y(\(a\))41 b Ft(a)255 2205 y Fm(0)322 2190 y Ft(<)28 b(a)477 2205 y Fm(1)544 2190 y Ft(<)g Fr(\001)17 b(\001)g(\001)26 b Ft(<)h(a)946 2205 y Fu(n)p Fv(\000)p Fm(1)1111 2190 y Ft(<)h(a)1266 2205 y Fu(n)1341 2190 y Fx(=)f Ft(a)1495 2205 y Fm(0)1557 2190 y Fx(+)22 b Ft(h)32 2306 y Fx(\(b\))42 b Ft(a)255 2321 y Fu(i)305 2306 y Fr(\000)23 b Ft(a)456 2321 y Fu(i)p Fv(\000)p Fm(1)602 2306 y Fr(\024)28 b Fx(min)o(\(2)957 2270 y Fv(\000)p Fu(l)1038 2306 y Ft(;)17 b Fx(2)1131 2270 y Fv(\000)p Fu(l)1211 2306 y Ft(=)-5 b(M)10 b Fx(\))33 b(for)f Ft(i)c Fx(=)f(1)p Ft(;)17 b(:)g(:)g(:)f(;)h (n)38 2450 y Fx(and)40 b(construct)i(an)f(appro)m(ximate)e(solution)g (on)i([)p Ft(a)1986 2465 y Fm(0)2026 2450 y Ft(;)17 b(a)2121 2465 y Fm(0)2188 2450 y Fx(+)27 b Ft(h)p Fx(];)45 b(similarly)37 b(this)j(can)h(b)s(e)g(done)g(on)38 2566 y([)p Ft(a)116 2581 y Fm(0)178 2566 y Fr(\000)22 b Ft(h;)17 b(a)428 2581 y Fm(0)468 2566 y Fx(].)137 2683 y(The)35 b(idea)d(is)h(to)g (start)g(at)g(\()p Ft(a)1207 2698 y Fm(0)1246 2683 y Ft(;)17 b(b)1331 2698 y Fm(0)1371 2683 y Fx(\))33 b(and)g(dra)m(w)h(a)f (line)f(with)h(slop)s(e)f Ft(f)11 b Fx(\()p Ft(a)2759 2698 y Fm(0)2799 2683 y Ft(;)17 b(b)2884 2698 y Fm(0)2923 2683 y Fx(\))33 b(un)m(til)f(it)g(in)m(tersects)38 2799 y Ft(x)d Fx(=)g Ft(a)278 2814 y Fm(1)318 2799 y Fx(,)k(sa)m(y)i(at)e (\()p Ft(a)757 2814 y Fm(1)796 2799 y Ft(;)17 b(b)881 2814 y Fm(1)921 2799 y Fx(\),)33 b(then)h(starting)f(from)f(\()p Ft(a)1927 2814 y Fm(1)1966 2799 y Ft(;)17 b(b)2051 2814 y Fm(1)2091 2799 y Fx(\))33 b(dra)m(w)h(a)f(line)f(with)h(slop)s(e)g Ft(f)11 b Fx(\()p Ft(a)3290 2814 y Fm(1)3329 2799 y Ft(;)17 b(b)3414 2814 y Fm(1)3454 2799 y Fx(\))33 b(un)m(til)38 2915 y(it)g(in)m(tersects)j Ft(x)31 b Fx(=)g Ft(a)815 2930 y Fm(2)854 2915 y Fx(,)k(sa)m(y)g(at)f(\()p Ft(a)1296 2930 y Fm(2)1336 2915 y Ft(;)17 b(b)1421 2930 y Fm(2)1460 2915 y Fx(\),)35 b(etc.)50 b(Since)34 b(w)m(e)i(w)m(an)m(t)f(an)f (appro)m(ximate)f(solution)g(whic)m(h)38 3031 y(maps)f(rationals)f(to)h (rationals,)f(w)m(e)i(appro)m(ximate)f(the)h(slop)s(es)f Ft(f)11 b Fx(\()p Ft(a)2551 3046 y Fu(i)p Fv(\000)p Fm(1)2670 3031 y Ft(;)17 b(b)2755 3046 y Fu(i)p Fv(\000)p Fm(1)2873 3031 y Fx(\))33 b(b)m(y)g(rationals)e Ft(s)3527 3046 y Fu(i)p Fv(\000)p Fm(1)3645 3031 y Fx(.)137 3147 y(More)i(precisely)-8 b(,)33 b(w)m(e)h(recursiv)m(ely)f(de\014ne)h(for)e Ft(i)c Fx(=)g(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(n)612 3305 y(')p Fx(\()p Ft(x)p Fx(\))28 b(=)g Ft(b)980 3320 y Fu(i)p Fv(\000)p Fm(1)1121 3305 y Fx(+)22 b(\()p Ft(x)g Fr(\000)h Ft(a)1485 3320 y Fu(i)p Fv(\000)p Fm(1)1604 3305 y Fx(\))p Ft(s)1688 3320 y Fu(i)p Fv(\000)p Fm(1)1903 3305 y Fx(for)32 b Ft(a)2103 3320 y Fu(i)p Fv(\000)p Fm(1)2250 3305 y Fr(\024)c Ft(x)g Fr(\024)g Ft(a)2594 3320 y Fu(i)2622 3305 y Fx(,)612 3456 y Ft(b)653 3471 y Fu(i)710 3456 y Fx(:=)f Ft(')p Fx(\()p Ft(a)993 3471 y Fu(i)1022 3456 y Fx(\))p Ft(;)635 3615 y Fr(\000)22 b Ft(M)39 b Fr(\024)28 b Ft(s)1018 3630 y Fu(i)p Fv(\000)p Fm(1)1164 3615 y Fr(\024)g Ft(M)108 b Fx(suc)m(h)34 b(that)f Fr(j)p Ft(s)1977 3630 y Fu(i)p Fv(\000)p Fm(1)2117 3615 y Fr(\000)22 b Ft(f)11 b Fx(\()p Ft(a)2364 3630 y Fu(i)p Fv(\000)p Fm(1)2483 3615 y Ft(;)17 b(b)2568 3630 y Fu(i)p Fv(\000)p Fm(1)2686 3615 y Fx(\))p Fr(j)27 b(\024)i Fx(2)2934 3579 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)3121 3615 y Fx(.)38 3787 y(Clearly)j Ft(')g Fx(is)g(con)m(tin)m(uous,)h(admissible)e(and)i(has)g(piecewise)g (deriv)-5 b(ativ)m(es)1203 3959 y Ft(')1267 3917 y Fv(0)1290 3959 y Fx(\()p Ft(x)p Fx(\))28 b(=)g Ft(s)1599 3974 y Fu(i)p Fv(\000)p Fm(1)1815 3959 y Fx(for)k Ft(a)2015 3974 y Fu(i)p Fv(\000)p Fm(1)2161 3959 y Ft(<)27 b(x)i(<)e(a)2502 3974 y Fu(i)2530 3959 y Fx(.)38 4130 y(No)m(w)33 b(for)f Ft(a)460 4145 y Fu(i)p Fv(\000)p Fm(1)606 4130 y Ft(<)c(x)g(<)f(a)947 4145 y Fu(i)1008 4130 y Fx(w)m(e)34 b(ha)m(v)m(e)g Fr(j)p Ft(x)22 b Fr(\000)h Ft(a)1633 4145 y Fu(i)p Fv(\000)p Fm(1)1751 4130 y Fr(j)28 b(\024)g Fx(2)1961 4094 y Fv(\000)p Fu(l)2074 4130 y Fx(and)827 4369 y Fr(j)p Ft(')p Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(b)1213 4384 y Fu(i)p Fv(\000)p Fm(1)1332 4369 y Fr(j)k(\024)h(j)p Ft(x)23 b Fr(\000)f Ft(a)1748 4384 y Fu(i)p Fv(\000)p Fm(1)1867 4369 y Fr(j)g(\001)f(j)p Ft(s)2040 4384 y Fu(i)p Fv(\000)p Fm(1)2158 4369 y Fr(j)28 b(\024)2329 4301 y Fx(2)2378 4265 y Fv(\000)p Fu(l)p 2329 4346 130 4 v 2341 4437 a Ft(M)2491 4369 y Fr(\001)21 b Ft(M)39 b Fx(=)27 b(2)2825 4327 y Fv(\000)p Fu(l)2906 4369 y Ft(;)38 4575 y Fx(hence)34 b(b)m(y)h(\(11\))360 4742 y Fr(j)p Ft(')452 4701 y Fv(0)475 4742 y Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))p Fr(j)27 b Fx(=)h Fr(j)p Ft(s)1390 4757 y Fu(i)p Fv(\000)p Fm(1)1530 4742 y Fr(\000)23 b Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))p Fr(j)1212 4893 y(\024)29 b(j)p Ft(s)1392 4908 y Fu(i)p Fv(\000)p Fm(1)1532 4893 y Fr(\000)22 b Ft(f)11 b Fx(\()p Ft(a)1779 4908 y Fu(i)p Fv(\000)p Fm(1)1897 4893 y Ft(;)17 b(b)1982 4908 y Fu(i)p Fv(\000)p Fm(1)2101 4893 y Fx(\))p Fr(j)22 b Fx(+)g Fr(j)p Ft(f)11 b Fx(\()p Ft(a)2463 4908 y Fu(i)p Fv(\000)p Fm(1)2581 4893 y Ft(;)17 b(b)2666 4908 y Fu(i)p Fv(\000)p Fm(1)2784 4893 y Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))p Fr(j)1212 5057 y(\024)29 b Fx(2)1367 5016 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1576 5057 y Fx(+)22 b(2)1723 5016 y Fv(\000)p Fu(k)r Fv(\000)p Fm(1)1938 5057 y Fx(=)28 b(2)2091 5016 y Fu(k)2133 5057 y Ft(:)38 5242 y Fx(Hence)34 b Ft(')e Fx(is)g(an)h(appro)m(ximate)e(solution)g(up)i(to)g(2)1900 5206 y Fv(\000)p Fu(k)1997 5242 y Fx(.)p 3682 5183 41 4 v 3682 5238 4 56 v 3720 5238 V 3682 5241 41 4 v 137 5426 a(The)g(appro)m(ximate)f(solutions)f(w)m(e)i(ha)m(v)m(e)g (constructed)h(are)e Fw(r)-5 b(ational)34 b(p)-5 b(olygons)8 b Fx(,)31 b(i.e.,)h(piecewise)38 5542 y(di\013eren)m(tiable)f(con)m (tin)m(uous)i(functions)g(with)f(rational)e(corners)k(and)e(rational)f (slop)s(es.)38 5686 y Fp(Lemma)51 b(7.3.)e Fw(Given)d(a)g(r)-5 b(ational)46 b(p)-5 b(olygon)46 b Ft(')h Fw(on)f Fx([)p Ft(a;)17 b(b)p Fx(])47 b Fw(and)f Ft(c)j Fr(2)h Fx([)p Ft(a;)17 b(b)p Fx(])p Fw(.)80 b(Then)46 b(one)g(of)g(the)38 5802 y(fol)5 b(lowing)33 b(alternatives)i(wil)5 b(l)34 b(hold.)38 5946 y Fx(\(a\))41 b(0)27 b Fr(\024)i Ft(')p Fx(\()p Ft(x)p Fx(\))35 b Fw(for)f Ft(a)28 b Fr(\024)g Ft(x)g Fr(\024)h Ft(b)p Fw(,)35 b(or)32 6062 y Fx(\(b\))42 b Ft(')p Fx(\()p Ft(x)p Fx(\))28 b Fr(\024)g Fx(0)35 b Fw(for)f Ft(a)28 b Fr(\024)g Ft(x)g Fr(\024)h Ft(b)p Fw(,)35 b(or)43 6179 y Fx(\(c\))42 b Fw(ther)-5 b(e)41 b(is)h(a)f Ft(d)e(<)h(c)h Fw(such)h(that)g Ft(')p Fx(\()p Ft(d)p Fx(\))d(=)h(0)h Fw(and)g(either)g Fx(0)e Fr(\024)i Ft(')p Fx(\()p Ft(x)p Fx(\))g Fw(for)h Ft(d)d Fr(\024)i Ft(x)f Fr(\024)g Ft(c)p Fw(,)j(or)e(else)204 6295 y Ft(')p Fx(\()p Ft(x)p Fx(\))28 b Fr(\024)g Fx(0)35 b Fw(for)f Ft(d)28 b Fr(\024)g Ft(x)g Fr(\024)g Ft(c)p Fw(.)p eop %%Page: 21 21 21 20 bop 38 450 a Fw(Pr)-5 b(o)g(of.)41 b Fx(Let)27 b Ft(a)h Fx(=)f Ft(a)738 465 y Fm(0)806 450 y Ft(<)g(a)960 465 y Fm(1)1028 450 y Ft(<)g Fr(\001)17 b(\001)g(\001)26 b Ft(<)i(a)1430 465 y Fu(n)1504 450 y Fx(=)g Ft(b)f Fx(b)s(e)g(the)g (exception)h(p)s(oin)m(ts)e(for)g Ft(')p Fx(.)42 b(W)-8 b(e)27 b(can)g(lo)s(cate)f Ft(c)h Fx(in)38 566 y Ft(a)h Fx(=)f Ft(a)271 581 y Fm(0)339 566 y Ft(<)g(a)493 581 y Fm(1)560 566 y Ft(<)h Fr(\001)17 b(\001)g(\001)26 b Ft(<)h(a)962 581 y Fu(n)1037 566 y Fx(=)h Ft(b)p Fx(.)43 b(Pic)m(k)30 b Ft(i)f Fx(maximal)e(suc)m(h)k(that)e Ft(a)2401 581 y Fu(i)p Fv(\000)p Fm(1)2547 566 y Fr(\024)f Ft(c)p Fx(.)43 b(Compare)29 b Ft(')p Fx(\()p Ft(a)3331 581 y Fm(0)3370 566 y Fx(\),)h Ft(')p Fx(\()p Ft(a)3618 581 y Fm(1)3658 566 y Fx(\),)38 683 y Ft(')p Fx(\()p Ft(a)191 698 y Fu(i)p Fv(\000)p Fm(1)309 683 y Fx(\))39 b(and)f Ft(')p Fx(\()p Ft(c)p Fx(\).)61 b(If)39 b(all)d(ha)m(v)m(e)k(the)f (same)f(sign,)i(w)m(e)f(are)g(done.)62 b(Otherwise)39 b(pic)m(k)g Ft(j)44 b Fx(maximal)38 799 y(suc)m(h)36 b(that)f Ft(')p Fx(\()p Ft(a)627 814 y Fu(j)664 799 y Fx(\))g(and)g Ft(')p Fx(\()p Ft(a)1082 814 y Fu(j)t Fm(+1)1209 799 y Fx(\))g(\(or)g Ft(')p Fx(\()p Ft(c)p Fx(\),)h(resp)s(ectiv)m (ely\))g(c)m(hange)g(sign.)51 b(Then)36 b(w)m(e)h(are)e(done)h(as)38 915 y(w)m(ell.)p 3682 856 41 4 v 3682 912 4 56 v 3720 912 V 3682 915 41 4 v 137 1084 a(W)-8 b(e)34 b(need)f(an)g(additional)d (restriction)i(in)g(order)h(to)f(estimate)g(the)h(di\013erence)g(of)g (appro)m(ximate)38 1201 y(solutions.)38 1338 y Fp(De\014nition)i(7.4.) 41 b Fx(A)32 b(function)f Ft(f)22 b Fx(:)33 b Ft(D)e Fr(!)c Fn(R)5 b Fx(,)38 b Ft(D)30 b Fr(\022)e Fn(R)2043 1302 y Fm(2)2120 1338 y Fx(is)k(said)f(to)g(satisfy)h(a)f Fw(Lipschitz)j(c)-5 b(ondition)38 1455 y Fx(w.r.t.)33 b(its)f(second)i(argumen)m(t)e(for)g(the)h(constan)m(t)h Ft(L)28 b(>)f Fx(0,)33 b(if)e(for)h(ev)m(ery)i(\()p Ft(x;)17 b(y)2882 1470 y Fm(1)2922 1455 y Fx(\))p Ft(;)g Fx(\()p Ft(x;)g(y)3189 1470 y Fm(2)3227 1455 y Fx(\))28 b Fr(2)g Ft(D)1182 1614 y Fr(j)p Ft(f)11 b Fx(\()p Ft(x;)17 b(y)1454 1629 y Fm(1)1493 1614 y Fx(\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(x;)17 b(y)1896 1629 y Fm(2)1935 1614 y Fx(\))p Fr(j)28 b(\024)g Ft(L)p Fr(j)p Ft(y)2276 1629 y Fm(1)2337 1614 y Fr(\000)23 b Ft(y)2485 1629 y Fm(2)2524 1614 y Fr(j)p Ft(:)38 1774 y Fp(Theorem)40 b(7.5)h Fx(\(F)-8 b(undamen)m(tal)34 b(Inequalit)m(y\))p Fp(.)43 b Fw(L)-5 b(et)38 b Ft(f)21 b Fx(:)35 b Ft(D)g Fr(!)d Fn(R)48 b Fw(b)-5 b(e)37 b(c)-5 b(ontinuous,)38 b(and)f(satisfy)g(a)38 1890 y Fs(Lipschitz)d Fw(c)-5 b(ondition)34 b(w.r.t.)h(its)g(se)-5 b(c)g(ond)34 b(ar)-5 b(gument)34 b(for)h(the)g(c)-5 b(onstant)34 b Ft(L)28 b(>)g Fx(0)p Fw(.)44 b(L)-5 b(et)1549 2049 y Ft(';)17 b( )e Fx(:)33 b([)p Ft(a;)17 b(b)p Fx(])28 b Fr(!)f Fn(R)38 2209 y Fw(b)-5 b(e)44 b(solutions)f(up)i(to)f Fx(2)906 2173 y Fv(\000)p Fu(k)1003 2209 y Ft(;)17 b Fx(2)1096 2173 y Fv(\000)p Fu(l)1221 2209 y Fw(of)64 b Fx(\(10\))p Fw(.)72 b(Assume)44 b Ft(')h Fr(\024)g Ft( )j Fw(on)c Ft(I)8 b Fw(,)46 b(or)e(else)g(that)g Ft(')g Fw(and)g Ft( )k Fw(ar)-5 b(e)38 2325 y(r)g(ational)34 b(p)-5 b(olygons.)44 b(Then)553 2461 y Fj(\014)553 2521 y(\014)586 2546 y Ft( )t Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)f Ft(')p Fx(\()p Ft(x)p Fx(\))1101 2461 y Fj(\014)1101 2521 y(\014)1162 2546 y Fr(\024)29 b Ft(e)1313 2505 y Fu(L)p Fm(\()p Fu(x)p Fv(\000)p Fu(a)p Fm(\))1552 2461 y Fj(\014)1552 2521 y(\014)1585 2546 y Ft( )t Fx(\()p Ft(a)p Fx(\))22 b Fr(\000)h Ft(')p Fx(\()p Ft(a)p Fx(\))2092 2461 y Fj(\014)2092 2521 y(\014)2147 2546 y Fx(+)2255 2478 y(2)2304 2442 y Fv(\000)p Fu(k)2424 2478 y Fx(+)f(2)2571 2442 y Fv(\000)p Fu(l)p 2255 2523 397 4 v 2420 2614 a Ft(L)2661 2465 y Fj(\000)2707 2546 y Ft(e)2752 2505 y Fu(L)p Fm(\()p Fu(x)p Fv(\000)p Fu(a)p Fm(\))3014 2546 y Fr(\000)g Fx(1)3162 2465 y Fj(\001)38 2741 y Fw(for)34 b(al)5 b(l)35 b Ft(x)28 b Fr(2)g Fx([)p Ft(a;)17 b(b)p Fx(])p Fw(.)38 2910 y(Pr)-5 b(o)g(of.)41 b(Part)46 b Fx(1.)55 b Ft(')35 b Fr(\024)g Ft( )40 b Fx(in)c Ft(I)8 b Fx(.)55 b(Let)37 b Ft(\033)h Fx(:=)d Ft( )29 b Fr(\000)c Ft(')p Fx(.)55 b(By)37 b(De\014nition)e(7.1)h(w)m(e)i(ha)m(v)m(e)g(for) e(all)e(p)s(oin)m(ts)38 3026 y(except)g(\014nitely)e(man)m(y)581 3101 y Fj(\014)581 3161 y(\014)614 3186 y Ft(')678 3144 y Fv(0)701 3186 y Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))1384 3101 y Fj(\014)1384 3161 y(\014)1444 3186 y Fr(\024)29 b Fx(2)1599 3144 y Fv(\000)p Fu(k)1793 3186 y Fx(and)2048 3101 y Fj(\014)2048 3161 y(\014)2081 3186 y Ft( )2148 3144 y Fv(0)2172 3186 y Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b( )t Fx(\()p Ft(x)p Fx(\)\))2857 3101 y Fj(\014)2857 3161 y(\014)2918 3186 y Fr(\024)28 b Fx(2)3072 3144 y Fv(\000)p Fu(l)3152 3186 y Ft(;)38 3354 y Fx(hence)34 b(with)e Ft(")27 b Fx(:=)h(2)784 3318 y Fv(\000)p Fu(k)903 3354 y Fx(+)22 b(2)1050 3318 y Fv(\000)p Fu(l)440 3514 y Ft( )507 3473 y Fv(0)530 3514 y Fx(\()p Ft(x)p Fx(\))g Fr(\000)h Ft(')847 3473 y Fv(0)870 3514 y Fx(\()p Ft(x)p Fx(\))28 b Fr(\024)1134 3429 y Fj(\014)1134 3489 y(\014)1168 3514 y Ft(f)11 b Fx(\()p Ft(x;)17 b( )t Fx(\()p Ft(x)p Fx(\)\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))2150 3429 y Fj(\014)2150 3489 y(\014)2206 3514 y Fx(+)22 b Ft(")27 b Fr(\024)h Ft(L)2548 3433 y Fj(\000)2594 3514 y Ft( )t Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)f Ft(')p Fx(\()p Ft(x)p Fx(\))3109 3433 y Fj(\001)3177 3514 y Fx(+)g Ft(")38 3673 y Fx(b)m(y)33 b(the)g Fs(Lipschitz)f Fx(condition,)f(hence)1482 3833 y Ft(\033)1541 3792 y Fv(0)1565 3833 y Fx(\()p Ft(x)p Fx(\))d Fr(\024)g Ft(L\033)t Fx(\()p Ft(x)p Fx(\))23 b(+)f Ft(":)38 3992 y Fx(Therefore)1007 4006 y Fj(Z)1107 4032 y Fu(x)1063 4231 y(a)1167 4141 y Ft(e)1212 4100 y Fv(\000)p Fu(Lt)1345 4061 y Fj(\000)1391 4141 y Ft(\033)1450 4100 y Fv(0)1473 4141 y Fx(\()p Ft(t)p Fx(\))g Fr(\000)h Ft(L\033)t Fx(\()p Ft(t)p Fx(\))1942 4061 y Fj(\001)2005 4141 y Ft(dt)k Fr(\024)i Ft(")2287 4006 y Fj(Z)2385 4032 y Fu(x)2341 4231 y(a)2446 4141 y Ft(e)2491 4100 y Fv(\000)p Fu(Lt)2640 4141 y Ft(dt:)38 4332 y Fx(The)34 b(in)m(tegrand)f(on)h(the) g(left-hand)e(side)i(has)f(\014nitely)g(man)m(y)h(discon)m(tin)m (uities)e(but)i(a)f(con)m(tin)m(uous)38 4448 y(inde\014nite)f(in)m (tegral,)f(so)1026 4570 y Fj(\002)1068 4650 y Ft(e)1113 4609 y Fv(\000)p Fu(Lt)1246 4650 y Ft(\033)t Fx(\()p Ft(t)p Fx(\))1416 4570 y Fj(\003)1457 4592 y Fu(x)1457 4690 y(a)1529 4650 y Fr(\024)d Ft(")1680 4570 y Fj(\002)1721 4650 y Fr(\000)1817 4583 y Fx(1)p 1808 4628 67 4 v 1808 4719 a Ft(L)1885 4650 y(e)1930 4609 y Fv(\000)p Fu(Lt)2063 4570 y Fj(\003)2104 4592 y Fu(x)2104 4690 y(a)1026 4871 y Ft(e)1071 4830 y Fv(\000)p Fu(Lx)1218 4871 y Ft(\033)t Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)g Ft(e)1576 4830 y Fv(\000)p Fu(La)1720 4871 y Ft(\033)t Fx(\()p Ft(a)p Fx(\))28 b Fr(\024)2060 4803 y Ft(")p 2049 4848 V 2049 4939 a(L)2126 4760 y Fj(\020)2185 4871 y Ft(e)2230 4830 y Fv(\000)p Fu(La)2397 4871 y Fr(\000)23 b Ft(e)2542 4830 y Fv(\000)p Fu(Lx)2688 4790 y Fj(\001)1026 5087 y Ft(\033)t Fx(\()p Ft(x)p Fx(\))28 b Fr(\024)h Ft(e)1395 5046 y Fu(L)p Fm(\()p Fu(x)p Fv(\000)p Fu(a)p Fm(\))1634 5087 y Ft(\033)t Fx(\()p Ft(a)p Fx(\))22 b(+)1961 5019 y Ft(")p 1950 5064 V 1950 5155 a(L)2027 5006 y Fj(\000)2072 5087 y Ft(e)2117 5046 y Fu(L)p Fm(\()p Fu(x)p Fv(\000)p Fu(a)p Fm(\))2379 5087 y Fr(\000)g Fx(1)2527 5006 y Fj(\001)2573 5087 y Ft(;)38 5276 y Fx(whic)m(h)33 b(is)f(the)h(required)g(inequalit) m(y)-8 b(.)137 5392 y Fw(Part)46 b Fx(2.)54 b Ft(')36 b Fx(and)g Ft( )k Fx(are)c(rational)e(p)s(olygons.)53 b(It)36 b(su\016ces)j(to)c(pro)m(v)m(e)j(the)e(claim)e(for)h(a)h (rational)38 5508 y Ft(c)p Fx(.)63 b(By)39 b(Lemma)f(7.3)h(w)m(e)h(can) f(assume)g Ft(')p Fx(\()p Ft(d)p Fx(\))g(=)f Ft( )t Fx(\()p Ft(d)p Fx(\))h(for)f(some)h Ft(d)g(<)f(c)p Fx(,)j(and)e Ft(')p Fx(\()p Ft(x)p Fx(\))g Fr(\024)g Ft( )t Fx(\()p Ft(x)p Fx(\))g(for)38 5625 y Ft(d)27 b Fr(\024)h Ft(x)g Fr(\024)h Ft(c)p Fx(.)43 b(Then)34 b(b)m(y)f(the)g(\014rst)g(part)1080 5845 y Ft( )t Fx(\()p Ft(c)p Fx(\))22 b Fr(\000)h Ft(')p Fx(\()p Ft(c)p Fx(\))k Fr(\024)1711 5778 y Fx(2)1760 5742 y Fv(\000)p Fu(k)1880 5778 y Fx(+)22 b(2)2027 5742 y Fv(\000)p Fu(l)p 1711 5822 397 4 v 1876 5914 a Ft(L)2117 5765 y Fj(\000)2163 5845 y Ft(e)2208 5804 y Fu(L)p Fm(\()p Fu(c)p Fv(\000)p Fu(d)p Fm(\))2459 5845 y Fr(\000)h Fx(1)2608 5765 y Fj(\001)2653 5845 y Ft(;)38 6034 y Fx(whic)m(h)33 b(is)f(an)g(ev)m(en)j(stronger)e(inequalit)m(y)-8 b(.)p 3682 5975 41 4 v 3682 6031 4 56 v 3720 6031 V 3682 6034 41 4 v 137 6204 a(T)g(o)31 b(pro)m(v)m(e)h(uniqueness)h(of)d(solutions) g(as)g(w)m(ell)g(as)h(the)g(existence)i(of)d(exact)h(solutions,)f(w)m (e)i(again)38 6320 y(need)h(the)g Fs(Lipschitz)f Fx(condition.)p eop %%Page: 22 22 22 21 bop 38 450 a Fp(Theorem)43 b(7.6)i Fx(\(Uniqueness\))p Fp(.)h Fw(L)-5 b(et)40 b Ft(f)22 b Fx(:)35 b Ft(D)40 b Fr(!)d Fn(R)52 b Fw(b)-5 b(e)39 b(c)-5 b(ontinuous,)41 b(and)f(satisfy)g(a)47 b Fs(Lipschitz)38 566 y Fw(c)-5 b(ondition)34 b(w.r.t.)g(its)h(se)-5 b(c)g(ond)34 b(ar)-5 b(gument.)45 b(L)-5 b(et)1618 726 y Ft(';)17 b( )e Fx(:)33 b Ft(I)j Fr(!)27 b Fn(R)38 887 y Fw(b)-5 b(e)35 b(two)g(\(exact\))g (solutions)g(of)57 b Fx(\(10\))o Fw(.)47 b(If)35 b Ft(')p Fx(\()p Ft(a)p Fx(\))28 b(=)h Ft( )t Fx(\()p Ft(a)p Fx(\))36 b Fw(for)f(some)g Ft(a)29 b Fr(2)g Ft(I)8 b Fw(,)35 b(then)h Ft(')p Fx(\()p Ft(x)p Fx(\))29 b(=)f Ft( )t Fx(\()p Ft(x)p Fx(\))36 b Fw(for)38 1003 y(al)5 b(l)34 b Ft(x)28 b Fr(2)g Ft(I)8 b Fw(.)38 1173 y(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)37 b(sho)m(w)h(that)e Ft(')p Fx(\()p Ft(x)p Fx(\))f(=)f Ft( )t Fx(\()p Ft(x)p Fx(\))j(for)f Ft(a)f Fr(\024)g Ft(x)g Fr(\024)g Ft(a)25 b Fx(+)g Ft(L=)p Fx(2,)37 b Ft(x)e Fr(2)g Ft(I)8 b Fx(.)56 b(Similarly)32 b(this)37 b(can)g(b)s(e)38 1289 y(sho)m(wn)d(for)e Ft(a)22 b Fr(\000)h Ft(L=)p Fx(2)k Fr(\024)h Ft(x)h Fr(\024)f Ft(a)p Fx(;)33 b(hence)h(the)f(claim)d(follo)m(ws.)137 1405 y(In)m(tegrating)i(the)h (t)m(w)m(o)h(equations)922 1565 y Ft(')986 1524 y Fv(0)1010 1565 y Fx(\()p Ft(x)p Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\))98 b(and)f Ft( )2120 1524 y Fv(0)2144 1565 y Fx(\()p Ft(x)p Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(x;)17 b( )t Fx(\()p Ft(x)p Fx(\)\))38 1725 y(w)m(e)33 b(obtain)f(from)f Ft(')p Fx(\()p Ft(a)p Fx(\))d(=)f Ft( )t Fx(\()p Ft(a)p Fx(\))937 1940 y Ft(')p Fx(\()p Ft(x)p Fx(\))c Fr(\000)f Ft( )t Fx(\()p Ft(x)p Fx(\))28 b(=)1584 1805 y Fj(Z)1684 1831 y Fu(x)1639 2030 y(a)1727 1860 y Fj(\000)1773 1940 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')p Fx(\()p Ft(t)p Fx(\)\))22 b Fr(\000)g Ft(f)11 b Fx(\()p Ft(t;)17 b( )t Fx(\()p Ft(t)p Fx(\)\))2675 1860 y Fj(\001)2737 1940 y Ft(dt)38 2152 y Fx(and)32 b(hence)406 2342 y Fr(j)p Ft(')p Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft( )t Fx(\()p Ft(x)p Fx(\))p Fr(j)28 b(\024)1110 2206 y Fj(Z)1209 2232 y Fu(x)1165 2432 y(a)1253 2257 y Fj(\014)1253 2317 y(\014)1286 2342 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')p Fx(\()p Ft(t)p Fx(\)\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(t;)17 b( )t Fx(\()p Ft(t)p Fx(\)\))2189 2257 y Fj(\014)2189 2317 y(\014)2238 2342 y Ft(dt)28 b Fr(\024)g Ft(L)2540 2206 y Fj(Z)2640 2232 y Fu(x)2595 2432 y(a)2683 2257 y Fj(\014)2683 2317 y(\014)2717 2342 y Ft(')p Fx(\()p Ft(t)p Fx(\))22 b Fr(\000)g Ft( )t Fx(\()p Ft(t)p Fx(\))3191 2257 y Fj(\014)3191 2317 y(\014)3241 2342 y Ft(dt:)38 2559 y Fx(Let)j Ft(M)35 b Fx(b)s(e)25 b(the)g(suprem)m(um)g(of)f(the)i(range)e(of)h Fr(j)p Ft(')6 b Fr(\000)g Ft( )t Fr(j)24 b Fx(on)h([)p Ft(a;)17 b(a)6 b Fx(+)g Ft(L=)p Fx(2].)41 b(Then)26 b(for)e Ft(a)k Fr(\024)g Ft(x)g Fr(\024)h Ft(a)6 b Fx(+)g Ft(L=)p Fx(2)1127 2767 y Fr(j)p Ft(')p Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft( )t Fx(\()p Ft(x)p Fx(\))p Fr(j)k(\024)h Ft(L)p Fx(\()p Ft(x)23 b Fr(\000)g Ft(a)p Fx(\))p Ft(M)38 b Fr(\024)2449 2700 y Fx(1)p 2449 2744 49 4 v 2449 2836 a(2)2507 2767 y Ft(M)5 b(;)38 2962 y Fx(hence)34 b Ft(M)k Fx(=)28 b(0)k(and)h (therefore)g Ft(')27 b Fx(=)h Ft( )37 b Fx(on)32 b([)p Ft(a;)17 b(a)22 b Fx(+)g Ft(L=)p Fx(2].)p 3682 2903 41 4 v 3682 2959 4 56 v 3720 2959 V 3682 2962 41 4 v 137 3132 a(The)34 b(example)38 3295 y(\(12\))1204 b Ft(y)1468 3254 y Fv(0)1518 3295 y Fx(=)27 b Ft(y)1673 3254 y Fm(1)p Fu(=)p Fm(3)1783 3295 y Ft(;)114 b(y)t Fx(\(0\))26 b(=)i Ft(y)2279 3310 y Fm(0)2318 3295 y Ft(:)38 3455 y Fx(sho)m(ws)36 b(that)e(the)h Fs(Lipschitz)e Fx(condition)g(is)h(indeed)h(necessary)i (for)c(uniqueness:)50 b(for)33 b Ft(y)3352 3470 y Fm(0)3422 3455 y Fx(=)e(0)j(w)m(e)38 3571 y(ha)m(v)m(e)g(t)m(w)m(o)f(solutions)f Ft(')p Fx(\()p Ft(x)p Fx(\))27 b(=)h(0)k(und)h Ft(')p Fx(\()p Ft(x)p Fx(\))28 b(=)g(\()1832 3532 y Fm(2)p 1832 3548 36 4 v 1832 3606 a(3)1877 3571 y Ft(x)p Fx(\))1970 3535 y Fm(3)p Fu(=)p Fm(2)2080 3571 y Fx(.)38 3711 y Fp(Theorem)g(7.7)h Fx(\(Exact)c(Solution\))p Fp(.)34 b Fw(L)-5 b(et)29 b Ft(f)21 b Fx(:)34 b Ft(D)c Fr(!)d Fn(R)39 b Fw(b)-5 b(e)27 b(c)-5 b(ontinuous,)29 b(and)f(satisfy)f(a)35 b Fs(Lipschitz)38 3827 y Fw(c)-5 b(ondition)37 b(w.r.t.)h(its)h(se)-5 b(c)g(ond)37 b(ar)-5 b(gument.)55 b(L)-5 b(et)39 b Fx(\()p Ft(a)1927 3842 y Fm(0)1967 3827 y Ft(;)17 b(b)2052 3842 y Fm(0)2091 3827 y Fx(\))34 b Fr(2)h Ft(D)41 b Fw(such)d(that)h(the)f (r)-5 b(e)g(ctangle)38 b Ft(R)i Fw(given)38 3943 y(by)g Fr(j)p Ft(x)27 b Fr(\000)f Ft(a)434 3958 y Fm(0)474 3943 y Fr(j)38 b(\024)g Ft(a)p Fw(,)k Fr(j)p Ft(y)29 b Fr(\000)e Ft(b)1028 3958 y Fm(0)1068 3943 y Fr(j)37 b(\024)i Ft(b)i Fw(is)f(in)g Ft(D)s Fw(.)61 b(Assume)41 b Fr(j)p Ft(f)11 b Fx(\()p Ft(x;)17 b(y)t Fx(\))p Fr(j)36 b(\024)i Ft(M)51 b Fw(for)41 b Fx(\()p Ft(x;)17 b(y)t Fx(\))37 b Fr(2)h Ft(R)q Fw(,)k(and)e(let)38 4059 y Ft(h)28 b Fx(:=)g(min)n(\()p Ft(a;)17 b(b=)-5 b(M)10 b Fx(\))p Fw(.)46 b(Then)34 b(we)h(c)-5 b(an)34 b(c)-5 b(onstruct)35 b(an)g(exact)g(solution)g Ft(')11 b Fx(:)33 b([)p Ft(a)2825 4074 y Fm(0)2887 4059 y Fr(\000)23 b Ft(h;)17 b(a)3138 4074 y Fm(0)3199 4059 y Fx(+)22 b Ft(h)p Fx(])29 b Fr(!)e Fn(R)46 b Fw(of)38 4175 y Fx(\(10\))34 b Fw(such)h(that)g Ft(')p Fx(\()p Ft(a)820 4190 y Fm(0)859 4175 y Fx(\))28 b(=)g Ft(b)1070 4190 y Fm(0)1110 4175 y Fw(.)38 4345 y(Pr)-5 b(o)g(of.)41 b Fx(By)46 b(Theorem)g(7.2)g(w)m(e)h(ha)m(v)m(e)g(an)e(appro)m(ximate)g (solution)f Ft(')2664 4360 y Fu(n)2757 4345 y Fx(up)i(to)g(2)3093 4309 y Fv(\000)p Fu(n)3194 4345 y Fx(,)j(whic)m(h)e(is)e(a)38 4462 y(rational)30 b(p)s(olygon,)i(o)m(v)m(er)h Ft(I)j Fx(:=)27 b([)p Ft(a)1292 4477 y Fm(0)1354 4462 y Fr(\000)c Ft(h;)17 b(a)1605 4477 y Fm(0)1666 4462 y Fx(+)22 b Ft(h)p Fx(].)137 4578 y(By)27 b(Lemma)e(6.4)h(the)g(sequence)j(\()p Ft(')1438 4593 y Fu(n)1485 4578 y Fx(\))1523 4593 y Fu(n)p Fv(2)p Fo(N)1691 4578 y Fx(uniformly)24 b(con)m(v)m(erges)29 b(o)m(v)m(er)e Ft(I)34 b Fx(to)26 b(a)g(con)m(tin)m(uous)h(func-)38 4694 y(tion)j Ft(')p Fx(.)43 b(Hence)33 b(the)e(sequence)j(\()p Ft(f)11 b Fx(\()p Ft(x;)17 b(')1527 4709 y Fu(n)1574 4694 y Fx(\()p Ft(x)p Fx(\)\)\))1781 4709 y Fu(n)1859 4694 y Fx(of)31 b(con)m(tin)m(uous)h(functions)f(uniformly)f(con)m(v)m (erges)38 4810 y(o)m(v)m(er)j Ft(I)41 b Fx(to)32 b(the)h(con)m(tin)m (uous)g(function)f Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\).)1943 4772 y Fm(2)2022 4810 y Fx(Therefore,)34 b(b)m(y)f(Theorem)g(6.5,)1060 5043 y(lim)1036 5103 y Fu(n)p Fv(!1)1236 4907 y Fj(Z)1336 4934 y Fu(b)1292 5133 y(a)1387 5043 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')1627 5058 y Fu(n)1673 5043 y Fx(\()p Ft(t)p Fx(\)\))g Ft(dt)28 b Fx(=)2056 4907 y Fj(Z)2156 4934 y Fu(b)2111 5133 y(a)2207 5043 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')p Fx(\()p Ft(t)p Fx(\)\))g Ft(dt:)137 5254 y Fx(W)-8 b(e)33 b(no)m(w)h(pro)m(v)m(e)f (that)g Ft(')f Fx(is)g(an)h(exact)g(solution.)42 b(By)33 b(the)g(c)m(hoice)g(of)f Ft(')2763 5269 y Fu(n)1285 5330 y Fj(\014)1285 5389 y(\014)1319 5414 y Ft(')1383 5373 y Fv(0)1383 5439 y Fu(n)1430 5414 y Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(x;)17 b(')1943 5429 y Fu(n)1989 5414 y Fx(\()p Ft(x)p Fx(\)\))2158 5330 y Fj(\014)2158 5389 y(\014)2219 5414 y Fr(\024)29 b Fx(2)2374 5373 y Fv(\000)p Fu(n)38 5576 y Fx(for)j(all)e Ft(x)e Fr(2)h Ft(I)40 b Fx(where)34 b Ft(')929 5540 y Fv(0)929 5601 y Fu(n)976 5576 y Fx(\()p Ft(x)p Fx(\))e(is)g(de\014ned.)45 b(In)m(tegrating)32 b(this)h(inequalit)m(y)e(from)g Ft(a)3039 5591 y Fm(0)3112 5576 y Fx(to)h Ft(x)h Fx(giv)m(es)796 5676 y Fj(\014)796 5736 y(\014)796 5796 y(\014)829 5655 y(Z)929 5682 y Fu(x)885 5881 y(a)922 5890 y Fh(0)973 5710 y Fj(\002)1014 5791 y Ft(')1078 5750 y Fv(0)1078 5816 y Fu(n)1125 5791 y Fx(\()p Ft(t)p Fx(\))22 b Fr(\000)h Ft(f)11 b Fx(\()p Ft(t;)17 b(')1598 5806 y Fu(n)1644 5791 y Fx(\()p Ft(t)p Fx(\)\))1793 5710 y Fj(\003)1852 5791 y Ft(dt)1938 5676 y Fj(\014)1938 5736 y(\014)1938 5796 y(\014)1998 5791 y Fr(\024)29 b Fx(2)2153 5750 y Fv(\000)p Fu(n)2254 5791 y Fx(\()p Ft(x)23 b Fr(\000)f Ft(a)2520 5806 y Fm(0)2560 5791 y Fx(\))28 b Fr(\024)g Fx(2)2780 5750 y Fv(\000)p Fu(n)2881 5791 y Ft(h:)p 38 5922 499 4 v 137 5989 a Fm(2)173 6021 y Fg(This)c(is)h(b)r(est)g(pro)n (v)n(ed)f(b)n(y)g(a)g(sligh)n(tly)h(more)f(general)f(setup,)j(where)e (metric)h(spaces)e(\(e.g.,)j Ff(R)3085 5991 y Fe(2)3128 6021 y Fg(\))f(are)f(considered.)38 6121 y(One)31 b(sho)n(ws)g(that)h (if)g Fd(')778 6133 y Fc(n)824 6121 y Fg(,)h Fd( )934 6133 y Fc(n)1011 6121 y Fg(are)e(uniformly)g(con)n(v)n(ergen)n(t)f(to)i Fd(')p Fg(,)h Fd( )s Fg(,)g(resp)r(ectiv)n(ely)-7 b(,)32 b(then)g(\()p Fd(')3038 6133 y Fc(n)3084 6121 y Fd(;)14 b( )3175 6133 y Fc(n)3220 6121 y Fg(\))32 b(is)g(uniformly)38 6220 y(con)n(v)n(ergen)n(t)23 b(to)i(\()p Fd(';)14 b( )s Fg(\),)26 b(and)g(if)f Fd(')1098 6232 y Fc(n)1169 6220 y Fg(is)g(uniformly)g(con)n(v)n(ergen)n(t)e(to)i Fd(')p Fg(,)h(then)g Fd(f)9 b Fg(\()p Fd(')2564 6232 y Fc(n)2609 6220 y Fg(\()p Fd(x)p Fg(\)\))27 b(is)e(uniformly)g(con)n(v)n(ergen)n (t)e(to)38 6320 y Fd(f)9 b Fg(\()p Fd(')p Fg(\()p Fd(x)p Fg(\)\).)p eop %%Page: 23 23 23 22 bop 38 450 a Fx(Since)33 b Ft(')357 465 y Fu(n)436 450 y Fx(is)f(con)m(tin)m(uous,)h(b)m(y)h(the)f(F)-8 b(undamen)m(tal)31 b(Theorem)i(5.4)f(of)g(Calculus)933 572 y Fj(\014)933 632 y(\014)933 691 y(\014)966 686 y Ft(')1030 701 y Fu(n)1077 686 y Fx(\()p Ft(x)p Fx(\))23 b Fr(\000)f Ft(')1394 701 y Fu(n)1441 686 y Fx(\()p Ft(a)1530 701 y Fm(0)1570 686 y Fx(\))g Fr(\000)1729 551 y Fj(Z)1829 577 y Fu(x)1785 776 y(a)1822 785 y Fh(0)1890 686 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')2130 701 y Fu(n)2176 686 y Fx(\()p Ft(t)p Fx(\)\))g Ft(dt)2428 572 y Fj(\014)2428 632 y(\014)2428 691 y(\014)2488 686 y Fr(\024)29 b Fx(2)2643 645 y Fv(\000)p Fu(n)2744 686 y Ft(h:)38 934 y Fx(Approac)m(hing)j(this)g(limit)d(giv)m (es)1117 1170 y Ft(')p Fx(\()p Ft(x)p Fx(\))22 b Fr(\000)h Ft(')p Fx(\()p Ft(a)1587 1185 y Fm(0)1626 1170 y Fx(\))f Fr(\000)1786 1034 y Fj(Z)1885 1061 y Fu(x)1841 1260 y(a)1878 1269 y Fh(0)1946 1170 y Ft(f)11 b Fx(\()p Ft(t;)17 b(')p Fx(\()p Ft(t)p Fx(\)\))g Ft(dt)27 b Fx(=)g(0)p Ft(:)38 1423 y Fx(Di\013eren)m(tiation)j(yields)i Ft(')1025 1387 y Fv(0)1048 1423 y Fx(\()p Ft(x)p Fx(\))c(=)g Ft(f)11 b Fx(\()p Ft(x;)17 b(')p Fx(\()p Ft(x)p Fx(\)\),)32 b(and)h Ft(')2053 1438 y Fu(n)2100 1423 y Fx(\()p Ft(a)2189 1438 y Fm(0)2228 1423 y Fx(\))28 b(=)g Ft(b)2439 1438 y Fm(0)2511 1423 y Fx(en)m(tails)k Ft(')p Fx(\()p Ft(a)2971 1438 y Fm(0)3010 1423 y Fx(\))c(=)f Ft(b)3220 1438 y Fm(0)3260 1423 y Fx(.)p 3682 1364 41 4 v 3682 1419 4 56 v 3720 1419 V 3682 1422 41 4 v 137 1625 a(F)-8 b(or)32 b(the)h(construction)g (ab)s(o)m(v)m(e)g(of)f(an)h(exact)g(solution)e(w)m(e)j(ha)m(v)m(e)g (made)e(use)h(of)f(the)h Fs(Lipschitz)38 1742 y Fx(condition.)49 b(Ho)m(w)m(ev)m(er,)38 b(it)c(is)g(w)m(ell)h(kno)m(wn)h(that)f (classically)e(one)i(has)g Fs(Peano)p Fx('s)h(existence)g(theo-)38 1858 y(rem,)c(whic)m(h)h(do)s(es)g(not)g(require)f(a)h Fs(Lipschitz)f Fx(condition.)137 1974 y(F)-8 b(ollo)m(wing)32 b([1,)j(7])g(w)m(e)h(no)m(w)f(w)m(an)m(t)h(to)e(argue)h(that)g Fs(Peano)p Fx('s)g(existence)h(theorem)f(en)m(tails)f(that)38 2090 y(for)g(ev)m(ery)j(real)e Ft(x)g Fx(w)m(e)h(can)g(decide)f (whether)i Ft(x)32 b Fr(\025)g Fx(0)j(or)g Ft(x)d Fr(\024)h Fx(0,)i(hence)i(w)m(e)f(cannot)f(exp)s(ect)i(to)e(b)s(e)38 2207 y(able)d(to)g(pro)m(v)m(e)i(it)d(constructiv)m(ely)-8 b(.)137 2323 y(Consider)27 b(again)d(the)i(initial)c(v)-5 b(alue)25 b(problem)f(\(12\).)41 b(First)25 b(note)g(that)h(the)g(con)m (tin)m(uous)g(function)38 2442 y Ft(f)11 b Fx(\()p Ft(x;)17 b(y)t Fx(\))29 b(:=)h Ft(y)538 2406 y Fm(1)p Fu(=)p Fm(3)682 2442 y Fx(satis\014es)35 b(a)f Fs(Lipschitz)f Fx(condition)g(w.r.t.)i (its)e(second)j(argumen)m(t,)e(for)g(0)c Ft(<)g(a)h(<)38 2558 y(y)k(<)c(b)36 b Fx(\(and)f(similarly)c(for)j Ft(b)f(<)e(y)k(<)d (a)g(<)g Fx(0\).)50 b(T)-8 b(o)35 b(see)h(this,)f(use)h(Lemma)e(4.2.)50 b(So)35 b(it)f(su\016ces)j(to)38 2677 y(\014nd)c(a)f(b)s(ound)h(on)g (the)g(deriv)-5 b(ativ)m(e)32 b(of)g Ft(y)1529 2641 y Fm(1)p Fu(=)p Fm(3)1638 2677 y Fx(.)43 b(But)33 b(this)f(is)g(easy)-8 b(,)34 b(since)2692 2638 y Fu(d)p 2673 2654 74 4 v 2673 2712 a(dy)2757 2677 y Ft(y)2809 2641 y Fm(1)p Fu(=)p Fm(3)2945 2677 y Fx(=)3059 2638 y Fm(1)p 3059 2654 36 4 v 3059 2712 a(3)3104 2677 y Ft(y)3156 2641 y Fv(\000)p Fm(2)p Fu(=)p Fm(3)3348 2677 y Ft(<)3461 2638 y Fm(1)p 3461 2654 V 3461 2712 a(3)3507 2677 y Ft(a)3558 2641 y Fv(\000)p Fm(2)p Fu(=)p Fm(3)38 2802 y Fx(for)e(0)27 b Ft(<)h(a)g(<)f(y)k(<)d(b)p Fx(.)137 2931 y(Therefore)k(in)d(case)i Ft(y)940 2946 y Fm(0)1007 2931 y Ft(>)c Fx(0)j(the)h(solution)d Ft(')1785 2946 y Fm(+)1844 2931 y Fx(\()p Ft(x)p Fx(\))g(:=)g(\()2182 2891 y Fm(2)p 2182 2908 V 2182 2965 a(3)2227 2931 y Ft(x)17 b Fx(+)g Ft(y)2444 2880 y Fm(2)p Fu(=)p Fm(3)2440 2955 y(0)2554 2931 y Fx(\))2592 2894 y Fm(3)p Fu(=)p Fm(2)2731 2931 y Fx(is)30 b(unique,)h(and)f(similarly)38 3059 y(in)d(case)i Ft(y)397 3074 y Fm(0)463 3059 y Ft(<)f Fx(0)f(the)i(solution)d Ft(')1235 3074 y Fv(\000)1294 3059 y Fx(\()p Ft(x)p Fx(\))i(:=)g Fr(\000)p Fx(\()1709 3020 y Fm(2)p 1709 3036 V 1709 3094 a(3)1754 3059 y Ft(x)12 b Fx(+)g Fr(j)p Ft(y)1985 3074 y Fm(0)2025 3059 y Fr(j)2053 3023 y Fm(2)p Fu(=)p Fm(3)2163 3059 y Fx(\))2201 3023 y Fm(3)p Fu(=)p Fm(2)2339 3059 y Fx(is)27 b(unique.)42 b(Pic)m(k)29 b Fr(j)p Ft(y)3075 3074 y Fm(0)3113 3059 y Fr(j)f Fx(small)d(enough)38 3188 y(suc)m(h)34 b(that)e(\()517 3149 y Fm(2)p 517 3165 V 517 3222 a(3)584 3188 y Fr(\000)23 b(j)p Ft(y)760 3203 y Fm(0)799 3188 y Fr(j)827 3152 y Fm(2)p Fu(=)p Fm(3)936 3188 y Fx(\))974 3152 y Fm(3)p Fu(=)p Fm(2)1112 3188 y Ft(>)1226 3149 y Fm(1)p 1226 3165 V 1226 3222 a(2)1271 3188 y Fx(.)137 3305 y(No)m(w)43 b(supp)s(ose)h(that)f(\(12\))e(has)i (a)g(solution)d Ft(')p Fx(,)45 b(for)d(a)h(giv)m(en)f(real)g Ft(y)2736 3320 y Fm(0)2775 3305 y Fx(.)73 b(Compare)42 b Ft(')p Fx(\(1\))g(with)38 3421 y([)p Fr(\000)p Fx(1)p Ft(=)p Fx(2)p Ft(;)17 b Fx(1)p Ft(=)p Fx(2].)56 b(If)37 b Ft(')p Fx(\(1\))e Ft(<)h Fx(1)p Ft(=)p Fx(2,)i(then)g Ft(y)1515 3436 y Fm(0)1589 3421 y Fr(6)p Ft(>)e Fx(0,)i(hence)g Ft(y)2138 3436 y Fm(0)2213 3421 y Fr(\024)e Fx(0.)57 b(If)37 b Fr(\000)p Fx(1)p Ft(=)p Fx(2)f Ft(<)f(')p Fx(\(1\),)j(then)g Ft(y)3461 3436 y Fm(0)3536 3421 y Fr(6)p Ft(<)d Fx(0,)38 3538 y(hence)f Ft(y)357 3553 y Fm(0)423 3538 y Fr(\025)29 b Fx(0.)809 3821 y(8.)48 b Fs(Rela)-7 b(ted)37 b(W)m(ork)i(on)f(Exa)n (ct)f(Real)g(Numbers)137 3995 y Fx(There)31 b(are)e(man)m(y)g(approac)m (hes)i(to)e(exact)h(real)e(n)m(um)m(b)s(er)i(computation)d(in)i(the)g (literature.)41 b(One)38 4111 y(of)34 b(those)i(-)e(using)i Fs(M)838 4103 y(\177)834 4111 y(obius)e Fx(\(or)h(linear)e (fractional\))f(transformations)h(-)i(has)g(b)s(een)h(put)f(forw)m(ard) 38 4228 y(b)m(y)d Fs(Ed)n(ala)-7 b(t)p Fx(;)32 b(a)d(go)s(o)s(d)g(surv) m(ey)k(can)d(b)s(e)g(found)h(in)e([8])h(\(see)h(also)e([9]\).)43 b(Exact)31 b(real)e(n)m(um)m(b)s(ers)i(based)38 4344 y(on)40 b(the)h(so-called)f(redundan)m(t)h Ft(b)p Fx(-adic)f(notation)g (ha)m(v)m(e)i(b)s(een)f(treated)g(b)m(y)i Fs(Wiedmer)d Fx([18])h(and)38 4460 y(b)m(y)32 b Fs(B)253 4452 y(\177)249 4460 y(ohm)f Fx(and)g Fs(Car)-7 b(twright)31 b Fx([6],)g(and)f(based)i (on)e(con)m(tin)m(ued)h(fractions)f(b)m(y)j Fs(Gosper)c Fx(\(1972\),)38 4576 y Fs(Vuillemin)35 b Fx([17],)g Fs(Nielsen)f Fx(and)h Fs(K)m(orner)n(up)f Fx([14])g(and)h(recen)m(tly)h(b)m(y)h Fs(Geuvers)c Fx(and)i Fs(Niqui)38 4693 y Fx([11)o(].)137 4809 y(Generally)-8 b(,)24 b(one)f(can)g(see)h(these)g(approac)m(hes)g (as)f(either)f(using)i Fs(Ca)n(uchy)h Fx(sequences)h(with)d(\(\014xed) 38 4925 y(or)37 b(separately)h(giv)m(en\))f(mo)s(dulus,)h(or)f(else)g Fs(Dedekind)g Fx(cuts.)59 b(W)-8 b(e)38 b(prefer)g Fs(Ca)n(uchy)i Fx(sequences)38 5041 y(o)m(v)m(er)48 b Fs(Dedekind)f Fx(cuts,)k(since)d(the)f(latter)g(are)g(giv)m(en)g(b)m(y)h(sets,)k(and) 47 b(hence)i(w)m(e)f(w)m(ould)f(need)38 5158 y(additional)34 b(en)m(umerating)j(devices)i(in)d(order)i(to)f(compute)g(appro)m (ximations)f(of)h(a)g(real)f(n)m(um)m(b)s(er)38 5274 y(presen)m(ted)47 b(as)e(a)f Fs(Dedekind)h Fx(cut.)80 b(There)46 b(is)e(not)h(m)m(uc)m(h)g(of)f(a)h(di\013erence)g(b)s(et)m (w)m(een)i(\014xed)f(or)38 5390 y(separate)38 b(mo)s(duli)d(for)j Fs(Ca)n(uchy)i Fx(sequences:)57 b(one)38 b(can)g(alw)m(a)m(ys)h (transform)e(one)h(form)e(in)m(to)h(the)38 5506 y(other.)43 b(Ho)m(w)m(ev)m(er,)35 b(in)c(order)h(to)g(k)m(eep)i(the)e(standard)g (series)h(represen)m(tations)g(of)f(particular)e(reals)38 5622 y(\(lik)m(e)i Ft(e)p Fx(\))g(w)m(e)i(prefer)f(to)f(w)m(ork)i(with) e(separate)h(mo)s(duli.)137 5739 y(A)d(recen)m(t)h(treatmen)m(t)e (\(including)e(an)i(implemen)m(tation)e(in)h Fs(Ma)-7 b(thema)g(tica)p Fx(\))30 b(has)g(b)s(een)g(giv)m(en)38 5855 y(b)m(y)51 b Fs(Andersson)d Fx(in)f(his)i(Master's)h(thesis)f([2]) f(\(based)i(on)e Fs(P)-9 b(almgren)p Fx('s)49 b([15]\).)91 b(He)49 b(treats)38 5971 y(trigonometric)21 b(functions,)26 b(and)f(includes)f Fs(Picard)p Fx('s)g(and)h Fs(Euler)p Fx('s)f(metho)s(ds)g(to)g(constructiv)m(ely)38 6087 y(pro)m(v)m(e)34 b(the)f(existence)h(of)e(solutions)f(for)h(di\013eren)m(tial)f (equations.)137 6204 y(Some)k(authors)h(\(in)e(particular)f(the)j (so-called)e(Russian)h(sc)m(ho)s(ol\))f(restrict)i(atten)m(tion)e(to)h (com-)38 6320 y(putable)h(real)g(n)m(um)m(b)s(ers.)56 b(W)-8 b(e)37 b(do)f(not)g(w)m(an)m(t)i(to)e(mak)m(e)h(this)f (restriction,)g(since)h(it)f(mak)m(es)h(sense,)p eop %%Page: 24 24 24 23 bop 38 450 a Fx(also)31 b(constructiv)m(ely)-8 b(,)33 b(to)f(sp)s(eak)h(ab)s(out)f(arbitrary)g(sequences.)46 b(This)33 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