%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: constr.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips constr.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.02:0934 %%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 0 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(ts)f(to)i(get)f(clear)f(ab)r(out)i(the)f(p)r(ossibilit)n(y)g(of)440 3853 y(suc)n(h)22 b(formalizations,)f(among)g(them)h(the)h(prosp)r(ect) e(to)h(use)g(formal)f(pro)r(ofs)g(of)h(existen)n(tial)440 3953 y(theorems)30 b(as)g(a)g(means)g(to)g(dev)n(elop)g(correct)f (programs,)g(in)h(this)h(case)f(in)n(v)n(olving)f(exact)440 4052 y(real)e(n)n(um)n(b)r(ers.)565 4152 y(F)-7 b(or)17 b(suc)n(h)i(purp)r(oses)e(it)i(is)g(of)f(imp)r(ortance)g(that)h(the)g (system)f(do)r(es)g(not)h(include)g(\(coun)n(t-)440 4252 y(able)34 b(or)f(dep)r(enden)n(t\))h(c)n(hoice)f(axioms.)55 b(The)34 b(aim)g(of)g(the)g(presen)n(t)f(pap)r(er)h(is)f(to)h(sk)n(etc) n(h)440 4351 y(the)h(dev)n(elopmen)n(t)g(of)f(elemen)n(tary)g (constructiv)n(e)g(analysis)g(on)g(this)h(basis,)i(up)e(to)g(and)440 4451 y(including)d(the)h(fundamen)n(tal)f(theorem)g(of)g(algebra.)48 b(The)32 b(k)n(ey)g(for)f(suc)n(h)h(a)g(c)n(hoice-free)440 4551 y(treatmen)n(t)d(is)g(to)f(pro)n(vide)g(witnesses)h(of)g (\(strictly\))g(p)r(ositiv)n(e)g(existen)n(tial)f(quan)n(ti\014ers)g (in)440 4650 y(all)f(de\014nitions.)440 4924 y Fx(2)135 b(Elemen)l(tary)47 b(Constructiv)l(e)f(Analysis)440 5106 y Fw(W)-7 b(e)24 b(understand)f(real)f(n)n(um)n(b)r(ers)h(as)f(mo)r (dulated)i Fv(Ca)n(uchy)h Fw(sequences)d(of)i(rationals.)3146 5076 y Fu(1)3217 5106 y Fw(So)440 5206 y(a)34 b Ft(r)l(e)l(al)j(numb)l (er,)f Fw(or)d(simply)i Ft(r)l(e)l(al)p Fw(,)h(is)f(nothing)f(but)h(a)f (sequence)g(\()p Fs(a)2641 5218 y Fr(m)2705 5206 y Fw(\))2737 5218 y Fr(m)p Fq(2)p Fp(N)2921 5206 y Fw(of)h(rational)p 440 5257 1146 4 v 532 5311 a Fo(1)567 5334 y Fn(Inasm)n(uc)n(h)28 b(as)g(w)n(e)f(do)h(not)h(fo)r(cus)e(on)h(\014xed)h(mo)r(duli,)e(w)n(e) g(deviate)i(from)d([2,)h(3,)g(21)q(].)42 b(An)28 b(approac)n(h)440 5413 y(similar)21 b(to)j(the)h(presen)n(t)f(one)g(w)n(as)g(c)n(hosen)h (in)e([22,)h(16].)1851 5662 y Fw(1)p eop %%Page: 2 2 2 1 bop 440 531 a Fw(n)n(um)n(b)r(ers)35 b(together)f(with)i(a)f(mo)r (dulus)h Fs(M)18 b Fw(:)30 b Fm(Q)1946 501 y Fu(+)2043 531 y Fl(!)36 b Fm(N)45 b Fw(suc)n(h)35 b(that)h Fl(j)p Fs(a)2707 543 y Fr(m)2794 531 y Fl(\000)23 b Fs(a)2926 543 y Fr(n)2971 531 y Fl(j)36 b(\024)g Fs(")f Fw(for)440 631 y Fs(m;)14 b(n)37 b Fl(\025)f Fs(M)9 b Fw(\()p Fs(")p Fw(\).)63 b(Let)36 b(us)g(stress)f(that)h(throughout)f(w)n(e)h (strictly)g(iden)n(tify)g(the)h(notions)440 731 y(\\real)21 b(n)n(um)n(b)r(er")g(and)h(\\mo)r(dulated)g Fv(Ca)n(uchy)i Fw(sequence)e(of)g(rationals")e(with)j(eac)n(h)e(other,)440 830 y(ev)n(en)30 b(though)f(w)n(e)h(sometimes)g(sa)n(y)f(that)h(an)g (instance)g(of)g(the)g(latter)g(represen)n(ts,)f(giv)n(es,)440 930 y(or)k(de\014nes)g(the)h(corresp)r(onding)e(instance)i(of)f(the)h (former.)54 b(As)34 b(usual,)h(w)n(e)e(denote)h(the)440 1029 y(set)542 999 y Fu(2)607 1029 y Fw(of)27 b(real)g(n)n(um)n(b)r (ers)g(b)n(y)g Fm(R)p Fw(.)565 1129 y(Ev)n(ery)20 b(rational)g Fs(q)25 b Fw(is)d(tacitly)g(understo)r(o)r(d)g(as)f(the)h(real)f (represen)n(ted)g(b)n(y)h(the)g(constan)n(t)440 1229 y(sequence)k Fs(a)827 1241 y Fr(m)913 1229 y Fw(=)c Fs(q)29 b Fw(with)e(the)f(constan)n(t)g(mo)r(dulus)g Fs(M)9 b Fw(\()p Fs(")p Fw(\))23 b(=)f(1.)36 b(F)-7 b(or)26 b(an)n(y)f(rational) g Fs(q)k Fw(with)440 1328 y Fl(j)p Fs(q)s Fl(j)44 b Fs(<)e Fw(1,)h(a)c(less)h(trivial)f(represen)n(tation)f(of)i(a)f(real)g(\(in)h (fact,)j(of)d(the)g(rational)3187 1296 y Fu(1)p 3145 1310 118 4 v 3145 1357 a(1)p Fq(\000)p Fr(q)3272 1328 y Fw(\))440 1445 y(is)g(that)f(giv)n(en)g(b)n(y)h(the)g(partial)e(sums) i Fs(a)1783 1457 y Fr(n)1871 1445 y Fw(=)1979 1383 y Fk(P)2066 1403 y Fr(n)2066 1470 y(i)p Fu(=0)2192 1445 y Fs(q)2232 1415 y Fr(i)2299 1445 y Fw(of)g(the)g(geometric)e(series:) 60 b(as)440 1545 y Fl(j)p Fs(a)507 1557 y Fr(m)592 1545 y Fl(\000)21 b Fs(a)722 1557 y Fr(n)767 1545 y Fl(j)31 b(\024)988 1512 y Fu(2)p 926 1526 157 4 v 926 1573 a Fq(j)p Fu(1)p Fq(\000)p Fr(q)r Fq(j)1093 1545 y Fl(j)p Fs(q)s Fl(j)1179 1515 y Fr(n)p Fu(+1)1341 1545 y Fw(whenev)n(er)g Fs(m)g Fl(\025)f Fs(n)p Fw(,)k(an)e(appropriate)e(mo)r(dulus)j(is)f Fs(M)9 b Fw(\()p Fs(")p Fw(\))31 b(=)440 1680 y(min)p Fl(f)14 b Fs(n)23 b Fl(2)g Fm(N)33 b Fl(j)23 b(j)p Fs(q)s Fl(j)1000 1650 y Fr(n)p Fu(+1)1153 1680 y Fl(\024)1250 1640 y Fq(j)p Fu(1)p Fq(\000)p Fr(q)r Fq(j)p 1250 1661 V 1312 1709 a Fu(2)1417 1680 y Fs(")14 b Fl(g)p Fw(.)565 1780 y(A)34 b(real)g Fs(a)g Fw(is)h Ft(nonne)l(gative)41 b Fw(\(written)35 b Fs(a)f Fl(2)h Fm(R)2018 1750 y Fu(0+)2112 1780 y Fw(\))g(if)g Fl(\000)p Fs(")f Fl(\024)g Fs(a)2543 1795 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))2734 1780 y Fw(for)34 b(all)g(rationals)440 1879 y Fs(")24 b(>)h Fw(0,)j(and)h Ft(p)l(ositive)36 b Fw(\(written)29 b Fs(a)c Fl(2)g Fm(R)1680 1849 y Fu(+)1741 1879 y Fw(\))k(if)g Fs(")c Fl(\024)f Fs(a)2076 1894 y Fr(M)6 b Fu(\()2181 1872 y Fj(")p 2181 1881 29 3 v 2181 1914 a Fi(2)2219 1894 y Fu(\))2278 1879 y Fw(for)28 b(some)g(rational)g Fs(")c(>)g Fw(0.)40 b(F)-7 b(or)440 1989 y(ev)n(ery)28 b(rational)h Fs(")d(>)g Fw(0)j(that)h (ensures)e Fs(a)e Fl(2)h Fm(R)1881 1959 y Fu(+)1972 1989 y Fw(as)i(in)h(the)g(de\014nition)g(thereof,)f(w)n(e)h(write)440 2088 y Fs(a)23 b Fl(2)562 2100 y Fr(")621 2088 y Fm(R)675 2058 y Fu(+)736 2088 y Fw(.)37 b(So)27 b(if)h Fs(a)23 b Fl(2)1109 2100 y Fr(")1168 2088 y Fm(R)1222 2058 y Fu(+)1283 2088 y Fw(,)28 b(then)g Fs(a)1567 2100 y Fr(m)1653 2088 y Fl(\025)1752 2056 y Fr(")p 1751 2070 34 4 v 1751 2117 a Fu(2)1821 2088 y Fw(for)g(all)f Fs(m)c Fl(\025)f Fs(M)9 b Fw(\()2380 2056 y Fr(")p 2379 2070 V 2379 2117 a Fu(2)2422 2088 y Fw(\).)565 2188 y(Giv)n(en)43 b(real)g(n)n(um)n(b)r (ers)g Fs(a;)14 b(b)43 b Fw(represen)n(ted)f(b)n(y)i(sequences)e(of)i (rationals)e(\()p Fs(a)3036 2200 y Fr(m)3099 2188 y Fw(\))3131 2200 y Fr(m)p Fq(2)p Fp(N)3281 2188 y Fw(,)440 2288 y(\()p Fs(b)508 2300 y Fr(m)571 2288 y Fw(\))603 2300 y Fr(m)p Fq(2)p Fp(N)782 2288 y Fw(with)29 b(mo)r(duli)h Fs(M)t(;)14 b(N)9 b Fw(,)29 b(w)n(e)f(de\014ne)h(eac)n(h)f Fs(c)h Fw(from)g(the)g(list)g Fs(a)19 b Fw(+)g Fs(b)p Fw(,)29 b Fl(\000)p Fs(a)p Fw(,)g Fl(j)p Fs(a)p Fl(j)p Fw(,)g Fs(a)19 b Fl(\001)h Fs(b)p Fw(,)440 2387 y(and)608 2355 y Fu(1)p 607 2369 37 4 v 607 2416 a Fr(a)676 2387 y Fw(\(the)k(latter)f (only)g(pro)n(vided)f(that)h Fl(j)p Fs(a)p Fl(j)h(2)1928 2399 y Fr(\021)1991 2387 y Fm(R)2045 2357 y Fu(+)2106 2387 y Fw(\))g(as)f(represen)n(ted)f(b)n(y)h(the)g(resp)r(ectiv)n(e)440 2487 y(sequence)k(\()p Fs(c)852 2499 y Fr(m)915 2487 y Fw(\))h(of)g(rationals)e(with)i(mo)r(dulus)g Fs(L)p Fw(:)694 2652 y Fs(c)p 1299 2682 4 100 v 621 w(c)1387 2664 y Fr(m)p 2005 2682 V 2057 2652 a Fs(L)p Fw(\()p Fs(")p Fw(\))p 644 2685 2457 4 v 694 2756 a Fs(a)18 b Fw(+)g Fs(b)p 1299 2786 4 101 v 476 w(a)1395 2768 y Fr(m)1476 2756 y Fw(+)g Fs(b)1595 2768 y Fr(m)p 2005 2786 V 2057 2756 a Fw(max)2225 2689 y Fk(\010)2274 2756 y Fs(M)2364 2689 y Fk(\000)2412 2723 y Fr(")p 2411 2737 34 4 v 2411 2785 a Fu(2)2454 2689 y Fk(\001)2492 2756 y Fs(;)c(N)2605 2689 y Fk(\000)2654 2723 y Fr(")p 2653 2737 V 2653 2785 a Fu(2)2696 2689 y Fk(\001\011)694 2856 y Fl(\000)p Fs(a)p 1299 2886 4 101 v 548 w Fl(\000)p Fs(a)1460 2868 y Fr(m)p 2005 2886 V 2057 2856 a Fs(M)2147 2789 y Fk(\000)2195 2824 y Fr(")p 2194 2838 34 4 v 2194 2885 a Fu(2)2237 2789 y Fk(\001)694 2956 y Fl(j)p Fs(a)p Fl(j)p 1299 2986 4 100 v 567 w(j)p Fs(a)1418 2968 y Fr(m)1481 2956 y Fl(j)p 2005 2986 V 553 w Fs(M)9 b Fw(\()p Fs(")p Fw(\))694 3081 y Fs(a)18 b Fl(\001)g Fs(b)p 1299 3135 4 150 v 518 w(a)1395 3093 y Fr(m)1476 3081 y Fl(\001)h Fs(b)1554 3093 y Fr(m)p 2005 3135 V 2057 3081 a Fw(max)2225 2989 y Fk(n)2280 3081 y Fs(M)2370 3014 y Fk(\000)2481 3049 y Fr(")p 2418 3063 158 4 v 2418 3110 a Fu(2)p Fr(K)2507 3121 y Fh(j)p Fj(b)p Fh(j)2585 3014 y Fk(\001)2623 3081 y Fs(;)14 b(N)2736 3014 y Fk(\000)2850 3049 y Fr(")p 2784 3063 164 4 v 2784 3110 a Fu(2)p Fr(K)2873 3121 y Fh(j)p Fj(a)p Fh(j)2957 3014 y Fk(\001)2995 2989 y(o)705 3248 y Fu(1)p 704 3262 37 4 v 704 3309 a Fr(a)777 3281 y Fw(for)27 b Fl(j)p Fs(a)p Fl(j)c(2)1072 3293 y Fr(\021)1136 3281 y Fm(R)1190 3250 y Fu(+)p 1299 3384 4 250 v 1351 3139 a Fk(\()1457 3191 y Fu(1)p 1428 3205 92 4 v 1428 3252 a Fr(a)1464 3260 y Fj(m)1612 3223 y Fw(if)28 b Fs(a)1732 3235 y Fr(m)1818 3223 y Fl(6)p Fw(=)23 b(0)1418 3344 y(0)152 b(if)28 b Fs(a)1732 3356 y Fr(m)1818 3344 y Fw(=)23 b(0)p 2005 3384 4 250 v 2057 3281 a(max)2225 3188 y Fk(n)2280 3281 y Fs(M)2370 3213 y Fk(\000)2418 3244 y Fr(\021)p 2418 3261 37 4 v 2420 3309 a Fu(2)2464 3213 y Fk(\001)2502 3281 y Fs(;)14 b(N)2615 3213 y Fk(\000)2663 3244 y Fr("\021)2730 3218 y Fi(2)p 2663 3261 101 4 v 2696 3309 a Fu(4)2773 3213 y Fk(\001)2811 3188 y(o)440 3521 y Fw(where)27 b(the)h(rational)f (n)n(um)n(b)r(er)g Fs(K)1506 3533 y Fr(a)1569 3521 y Fw(=)c(max)p Fl(f)14 b Fs(a)1912 3533 y Fr(n)1980 3521 y Fl(j)23 b Fs(n)g Fl(\024)g Fs(M)9 b Fw(\(1\))14 b Fl(g)k Fw(+)g(1)27 b(is)h(so)f(that)h Fs(a)3019 3533 y Fr(m)3106 3521 y Fl(\024)23 b Fs(K)3265 3533 y Fr(a)440 3621 y Fw(for)j(all)h Fs(m)c Fl(2)g Fm(N)t Fw(.)43 b(Needless)26 b(to)h(sa)n(y)-7 b(,)25 b Fs(a)17 b Fl(\000)f Fs(b)27 b Fw(and)1943 3588 y Fr(a)p 1943 3602 37 4 v 1946 3650 a(b)2016 3621 y Fw(\(for)f Fs(b)d Fl(2)g Fm(R)2365 3591 y Fu(+)2426 3621 y Fw(\))k(stand)g(for)f Fs(a)17 b Fw(+)f(\()p Fl(\000)p Fs(b)p Fw(\))27 b(and)440 3721 y Fs(a)18 b Fl(\001)554 3688 y Fu(1)p 554 3702 34 4 v 556 3749 a Fr(b)597 3721 y Fw(,)28 b(resp)r(ectiv)n(ely)-7 b(.)565 3820 y(W)g(e)36 b(write)g Fs(a)g Fl(\024)h Fs(b)f Fw(for)f Fs(b)24 b Fl(\000)f Fs(a)37 b Fl(2)g Fm(R)1701 3790 y Fu(0+)1832 3820 y Fw(and)e Fs(a)i(<)g(b)e Fw(for)h Fs(b)23 b Fl(\000)h Fs(a)37 b Fl(2)g Fm(R)2766 3790 y Fu(+)2827 3820 y Fw(.)62 b(Un)n(winding)440 3920 y(the)34 b(de\014nitions)f (yields)h(that)g Fs(a)e Fl(\024)h Fs(b)g Fw(is)g(to)h(sa)n(y)e(that)i (for)f(ev)n(ery)f(rational)g Fs(\021)k(>)c Fw(0)i(there)440 4019 y(exists)e Fs(P)727 4031 y Fr(\021)800 4019 y Fl(2)g Fm(N)42 b Fw(suc)n(h)33 b(that)g Fs(a)1401 4031 y Fr(m)1495 4019 y Fl(\024)e Fs(b)1627 4031 y Fr(m)1712 4019 y Fw(+)21 b Fs(\021)36 b Fw(for)c(all)h Fs(m)e Fl(\025)g Fs(P)2381 4031 y Fr(\021)2422 4019 y Fw(.)52 b(F)-7 b(urthermore,)33 b Fs(a)f(<)f(b)h Fw(is)440 4119 y(a)d(shorthand)g(for)g(the)h(presence) f(of)h(a)f(rational)f Fs(\021)i(>)c Fw(0)j(and)g(a)h(p)r(ositiv)n(e)f (in)n(teger)g Fs(Q)g Fw(with)440 4219 y Fs(a)484 4231 y Fr(m)568 4219 y Fw(+)20 b Fs(\021)32 b Fl(\024)c Fs(b)855 4231 y Fr(m)949 4219 y Fw(for)i(all)h Fs(m)e Fl(\025)f Fs(Q)p Fw(;)k(w)n(e)f(then)g(write)g Fs(a)e(<)2186 4231 y Fr(\021)r(;Q)2326 4219 y Fs(b)i Fw(\(or)f(simply)h Fs(a)e(<)2937 4231 y Fr(\021)3006 4219 y Fs(b)h Fw(if)i Fs(Q)e Fw(is)440 4318 y(not)e(needed\))g(whenev)n(er)e(w)n(e)h(w)n(an)n (t)g(to)h(call)f(these)h(witnesses.)565 4418 y(Constructiv)n(ely)-7 b(,)33 b(equalit)n(y)f Fs(a)g Fw(=)f Fs(b)i Fw(and)g(inequalit)n(y)f Fs(a)g Fl(6)p Fw(=)f Fs(b)i Fw(b)r(et)n(w)n(een)g(real)f(n)n(um)n(b)r (ers)440 4518 y(are)22 b(to)h(b)r(e)h(de\014ned)f(as)g Fs(a)g Fl(\025)g Fs(b)g Fl(^)h Fs(a)f Fl(\024)f Fs(b)h Fw(and)g Fs(a)g(<)g(b)g Fl(_)h Fs(a)f(>)f(b)p Fw(,)i(resp)r(ectiv)n (ely)-7 b(.)35 b(Some)23 b(routine)440 4617 y(pro)r(ofs)33 b(sho)n(w)g(that)h(the)g(arithmetic)g(of)g(real)f(n)n(um)n(b)r(ers)g (resp)r(ects)g(equalit)n(y)-7 b(.)55 b(Moreo)n(v)n(er,)440 4717 y Fs(a)23 b Fl(\024)g Fs(b)h Fw(could)h(equiv)-5 b(alen)n(tly)25 b(b)r(e)g(de\014ned)h(as)e Fl(:)p Fw(\()p Fs(a)g(>)e(b)p Fw(\),)k(and)f Fs(a)e Fw(=)g Fs(b)h Fw(as)h Fl(:)p Fw(\()p Fs(a)e Fl(6)p Fw(=)g Fs(b)p Fw(\),)j(whereas)440 4816 y Fs(a)41 b(<)g(b)d Fw(and)h Fs(a)i Fl(6)p Fw(=)g Fs(b)d Fw(are)g(constructiv)n(ely)f(stronger)g(than)i Fl(:)p Fw(\()p Fs(a)j Fl(\025)f Fs(b)p Fw(\))d(and)g Fl(:)p Fw(\()p Fs(a)k Fw(=)f Fs(b)p Fw(\),)440 4916 y(resp)r(ectiv)n (ely)-7 b(.)882 4886 y Fu(3)960 4916 y Fw(In)n(terv)i(als)29 b(are)f(then)i(de\014ned)f(in)h(the)f(familiar)g(w)n(a)n(y)-7 b(,)29 b(so)f([)p Fs(a;)14 b(b)p Fw(])29 b(as)g(the)g(set)440 5016 y Fl(f)14 b Fs(c)22 b Fl(2)i Fm(R)29 b Fl(j)23 b Fs(a)g Fl(\024)f Fs(c)h Fl(\024)g Fs(b)14 b Fl(g)p Fw(.)p 440 5069 1146 4 v 532 5123 a Fo(2)567 5146 y Fn(As)19 b(w)n(e)h(hesitate)h(to)f(dev)n(ote)h(ourselv)n(es)f(to)g(a)g(set)g (theory)g(whatso)r(ev)n(er,)h(w)n(e)f(wish)f(to)h(understand)h(eac)n(h) 440 5225 y(set)j(iden)n(ti\014ed)h(with)f(the)g(de\014ning)h(prop)r (ert)n(y)f(of)f(its)h(elemen)n(ts.)532 5281 y Fo(3)567 5304 y Fn(This)d(is)g(clear)h(in)f(view)h(of)f(the)i(existen)n(tial)f (and)h(univ)n(ersal)e(c)n(haracter)i(of)e(the)i(predicates)g Fg(2)c Ff(R)3118 5281 y Fo(+)3191 5304 y Fn(and)440 5383 y Fg(2)h Ff(R)555 5360 y Fo(0+)636 5383 y Fn(,)j(resp)r(ectiv)n(ely)-6 b(.)1851 5662 y Fw(2)p eop %%Page: 3 3 3 2 bop 565 531 a Fw(Except)25 b(for)h(the)g(completeness)g(prop)r (erties)f(whic)n(h)h(w)n(e)g(shall)g(discuss)g(subsequen)n(tly)-7 b(,)440 631 y(it)29 b(is)g(routine)f(to)g(v)n(erify)g(that)h(the)g (reals)f(as)g(de\014ned)h(ab)r(o)n(v)n(e)e(p)r(ossess)h(all)g(the)h (features)f(of)440 731 y(the)i(reals)f(that)i(are)e(commonly)g (considered)g(to)h(b)r(e)g(constructiv)n(e)f([4].)44 b(Constructiv)n(ely)-7 b(,)440 830 y(in)37 b(particular,)h(w)n(e)e (cannot)g(compare)g(t)n(w)n(o)f(reals,)j(but)g(can)e(compare)f(eac)n(h) h(real)g(with)440 930 y(ev)n(ery)26 b(non)n(trivial)h(in)n(terv)-5 b(al,)27 b(as)g(in)h(the)g(follo)n(wing)e Ft(appr)l(oximate)31 b(splitting)g(principle)p Fw(.)440 1074 y Fe(Lemma)f(1.)40 b Ft(F)-6 b(or)30 b(al)t(l)h(r)l(e)l(al)f(numb)l(ers)f Fs(a;)14 b(b;)g(c)p Ft(,)29 b(if)i Fs(a)23 b(<)g(b)p Ft(,)29 b(then)h(either)h Fs(a)23 b(<)f(c)30 b Ft(or)g Fs(c)23 b(<)g(b)p Ft(.)440 1219 y(Pr)l(o)l(of.)43 b Fw(Let)32 b Fs(M)t(;)14 b(N)t(;)g(L)32 b Fw(denote)g(the)h(mo)r(duli)f(of)g Fs(a;)14 b(b;)g(c)p Fw(,)33 b(resp)r(ectiv)n(ely)-7 b(,)33 b(and)f(assume)f(that)440 1319 y Fs(a)23 b(<)572 1331 y Fu(7)p Fr(\021)r(;P)739 1319 y Fs(b)p Fw(,)k(i.e.)1324 1418 y Fs(a)1368 1430 y Fr(m)1450 1418 y Fw(+)18 b(3)p Fs(\021)26 b(<)c(a)1773 1430 y Fr(m)1855 1418 y Fw(+)c(4)p Fs(\021)25 b Fl(\024)e Fs(b)2170 1430 y Fr(m)2251 1418 y Fl(\000)18 b Fw(3)p Fs(\021)440 1552 y Fw(for)35 b(ev)n(ery)f Fs(m)i Fl(\025)f Fs(P)12 b Fw(.)60 b(F)-7 b(or)35 b Fs(Q)g Fw(=)h(max)o Fl(f)p Fs(M)9 b Fw(\()p Fs(\021)s Fw(\))p Fs(;)14 b(N)9 b Fw(\()p Fs(\021)s Fw(\))p Fs(;)14 b(L)p Fw(\()p Fs(\021)s Fw(\))p Fs(;)g(P)e Fl(g)p Fw(,)37 b(w)n(e)e(either)g (ha)n(v)n(e)f Fs(c)3147 1564 y Fr(Q)3240 1552 y Fs(<)440 1651 y(a)484 1663 y Fr(Q)560 1651 y Fw(+)18 b(4)p Fs(\021)32 b Fw(or)c(else)g Fs(a)1063 1663 y Fr(Q)1139 1651 y Fw(+)18 b(4)p Fs(\021)28 b Fl(\024)c Fs(c)1458 1663 y Fr(Q)1515 1651 y Fw(,)29 b(but)g(not)g(b)r(oth.)41 b(In)29 b(the)g(former)f (case,)g Fs(c)2853 1663 y Fr(Q)2934 1651 y Fs(<)d(b)3060 1663 y Fr(Q)3135 1651 y Fl(\000)19 b Fw(3)p Fs(\021)440 1751 y Fw(and)27 b(th)n(us)1276 1851 y Fs(c)1312 1863 y Fr(m)1393 1851 y Fw(+)18 b Fs(\021)27 b Fl(\024)22 b Fs(c)1667 1863 y Fr(Q)1742 1851 y Fw(+)c(2)p Fs(\021)26 b(<)c(b)2057 1863 y Fr(Q)2132 1851 y Fl(\000)c Fs(\021)26 b Fl(\024)d Fs(b)2406 1863 y Fr(m)440 1984 y Fw(for)h(ev)n(ery)g Fs(m)f Fl(\025)f Fs(Q)p Fw(,)j(whic)n(h)g(is)g(to)g(sa)n(y)e(that)i Fs(c)e(<)1934 1996 y Fr(\021)r(;Q)2069 1984 y Fs(b)p Fw(.)36 b(In)25 b(the)g(latter)f(case,)h Fs(a)2869 1996 y Fr(Q)2938 1984 y Fw(+)13 b(3)p Fs(\021)25 b(<)e(c)3248 1996 y Fr(Q)440 2084 y Fw(and)k(th)n(us)1268 2183 y Fs(a)1312 2195 y Fr(m)1393 2183 y Fw(+)18 b Fs(\021)26 b Fl(\024)d Fs(a)1675 2195 y Fr(Q)1750 2183 y Fw(+)18 b(2)p Fs(\021)25 b(<)e(c)2065 2195 y Fr(Q)2140 2183 y Fl(\000)18 b Fs(\021)26 b Fl(\024)d Fs(c)2414 2195 y Fr(m)440 2317 y Fw(for)k(ev)n(ery)f Fs(m)d Fl(\025)g Fs(Q)p Fw(,)k(whic)n(h)h(is)f(to)h(sa)n(y)e(that)i Fs(a)23 b(<)1964 2329 y Fr(\021)r(;Q)2099 2317 y Fs(c)p Fw(.)p 3244 2317 4 57 v 3248 2264 50 4 v 3248 2317 V 3297 2317 4 57 v 565 2477 a(It)36 b(is)g(notew)n(orth)n(y)f(that)i (although)f Fs(a)h(<)g(c)g Fw(and)f Fs(c)h(<)g(b)f Fw(ma)n(y)g(w)n(ell) g(happ)r(en)h(sim)n(ul-)440 2577 y(taneously)-7 b(,)40 b(the)f(pro)r(of)f(of)h(Lemma)f(1)g(con)n(tains)f(a)h(routine)g(that)h (in)g(ev)n(ery)e(suc)n(h)h(case)440 2677 y(automatically)31 b(pins)i(do)n(wn)e(one)h(of)h(the)f(alternativ)n(es.)50 b(This)32 b(virtue,)h(whic)n(h)f(of)h(course)440 2776 y(stands)e(and)g(falls)g(with)g(the)h(presence)e(of)h(the)h(mo)r(duli)f (for)g Fs(a;)14 b(b;)g(c)30 b Fw(and)h(the)h(witness)f(for)440 2876 y Fs(a)23 b(<)g(b)p Fw(,)j(is)h(of)g(particular)e(relev)-5 b(ance)26 b(whenev)n(er)g(the)h(appro)n(ximate)e(splitting)i(principle) f(is)440 2975 y(applied)32 b(in\014nitely)g(often,)i(e.g.)d(in)h(the)g (pro)r(of)g(of)f(Theorem)g(12)g(b)r(elo)n(w.)50 b(By)31 b(emplo)n(ying)440 3075 y(that)d(routine,)f(namely)-7 b(,)27 b(w)n(e)h(can)f(a)n(v)n(oid)f(the)i(usual)f(in)n(v)n(o)r(cation) f(of)i(dep)r(enden)n(t)g(c)n(hoice)e(to)440 3175 y(successiv)n(ely)g (construct)h(a)g(sequence)g(of)h(real)f(n)n(um)n(b)r(ers.)565 3274 y(An)33 b(immediate)f(consequence)g(of)g(Lemma)h(1)f(is)g(that)h Fs(b)e Fl(\024)g Fs(c)i Fw(is)f(to)g(sa)n(y)g(that)h Fs(a)e(<)g(b)440 3374 y Fw(implies)d Fs(a)23 b(<)f(c)28 b Fw(for)f(all)g(reals)g Fs(c)p Fw(.)565 3474 y(F)-7 b(or)32 b(1)g Fl(\024)g Fs(j)38 b Fl(\024)32 b Fs(n)h Fw(let)g Fs(a)1311 3486 y Fr(j)1379 3474 y Fw(b)r(e)h(a)f(real)f(n)n (um)n(b)r(er)h(giv)n(en)g(b)n(y)f(the)i(sequence)f(of)g(rationals)440 3573 y(\()p Fs(a)516 3585 y Fr(j)s(m)610 3573 y Fw(\))642 3585 y Fr(m)p Fq(2)p Fp(N)822 3573 y Fw(with)d(mo)r(dulus)g Fs(M)1431 3585 y Fr(j)1465 3573 y Fw(.)43 b(Then)29 b(the)h Ft(maximum)36 b Fw(max)o Fl(f)14 b Fs(a)2539 3585 y Fr(j)2600 3573 y Fl(j)26 b Fs(j)31 b Fl(\024)26 b Fs(n)14 b Fl(g)29 b Fw(is)g(the)h(real)440 3673 y(n)n(um)n(b)r(er)d(represen)n(ted)e(b)n (y)i(the)h(sequence)e(of)h(rationals)2218 3606 y Fk(\000)2256 3673 y Fw(max)p Fl(f)14 b Fs(a)2511 3685 y Fr(j)s(m)2627 3673 y Fl(j)23 b Fs(j)28 b Fl(\024)23 b Fs(n)14 b Fl(g)2929 3606 y Fk(\001)2966 3706 y Fr(m)p Fq(2)p Fp(N)3143 3673 y Fw(with)440 3781 y(mo)r(dulus)36 b(max)p Fl(f)14 b Fs(M)1075 3793 y Fr(j)1146 3781 y Fl(j)37 b Fs(j)43 b Fl(\024)37 b Fs(n)14 b Fl(g)p Fw(.)61 b(Constructiv)n(ely)-7 b(,)38 b(one)d(cannot)h(pic)n(k)g(an)g(index)g Fs(i)g Fw(for)440 3881 y(whic)n(h)30 b Fs(a)724 3893 y Fr(i)778 3881 y Fw(=)c(max)o Fl(f)14 b Fs(a)1123 3893 y Fr(j)1184 3881 y Fl(j)27 b Fs(j)k Fl(\024)26 b Fs(n)14 b Fl(g)p Fw(,)30 b(but)g(for)f(eac)n(h)g Fs(")d(>)g Fw(0)k(w)n(e)f(can)g(\014nd) h({)g(as)f(follo)n(ws)f({)i(an)440 3980 y(index)e Fs(i)f Fw(suc)n(h)g(that)h Fs(a)1131 3992 y Fr(i)1186 3980 y Fw(is)g(maximal)f Ft(up)j(to)i Fs(")c Fw(among)e(all)i(the)g Fs(a)2465 3992 y Fr(j)2500 3980 y Fw(.)440 4125 y Fe(Lemma)f(2.)39 b Ft(F)-6 b(or)28 b(al)t(l)h(r)l(e)l(al)f(numb)l(ers)f Fs(a)1688 4137 y Fu(1)1725 4125 y Fs(;)14 b(:)g(:)g(:)g(;)g(a)1954 4137 y Fr(n)2027 4125 y Ft(and)28 b(e)l(ach)h(r)l(ational)g Fs(")23 b(>)f Fw(0)27 b Ft(we)i(c)l(an)e(\014nd)440 4225 y(an)j Fs(i)22 b Fl(\024)h Fs(n)30 b Ft(so)g(that)f Fs(a)1097 4237 y Fr(j)1155 4225 y Fl(\024)23 b Fs(a)1287 4237 y Fr(i)1333 4225 y Fw(+)18 b Fs(")29 b Ft(for)i(every)g Fs(j)d Fl(\024)22 b Fs(n)p Ft(.)440 4369 y(Pr)l(o)l(of.)43 b Fw(F)-7 b(or)45 b(eac)n(h)f Fs(j)57 b Fl(\024)51 b Fs(n)45 b Fw(let)g Fs(a)1554 4381 y Fr(j)1634 4369 y Fw(b)r(e)h(represen)n(ted)e(b)n(y)g(the)i(sequence)e(of)h(rationals)440 4469 y(\()p Fs(a)516 4481 y Fr(j)s(m)610 4469 y Fw(\))642 4481 y Fr(m)p Fq(2)p Fp(N)820 4469 y Fw(with)28 b(mo)r(dulus)g Fs(M)1425 4481 y Fr(j)1459 4469 y Fw(.)37 b(Giv)n(en)28 b(a)f(rational)f Fs(")d(>)g Fw(0,)k(set)893 4649 y Fs(M)k Fw(=)23 b(max)p Fl(f)14 b Fs(M)1385 4661 y Fr(j)1418 4582 y Fk(\000)1468 4593 y Fs(")p 1466 4630 42 4 v 1466 4706 a Fw(2)1518 4582 y Fk(\001)1579 4649 y Fl(j)23 b Fs(j)28 b Fl(\024)23 b Fs(n)14 b Fl(g)p Fs(;)96 b(b)23 b Fw(=)f(max)p Fl(f)14 b Fs(a)2401 4661 y Fr(j)s(M)2528 4649 y Fl(j)23 b Fs(j)28 b Fl(\024)22 b Fs(n)14 b Fl(g)p Fs(;)440 4828 y Fw(and)1371 4928 y Fs(i)23 b Fw(=)g(min)p Fl(f)14 b Fs(k)25 b Fl(\024)e Fs(n)g Fl(j)g Fs(a)2024 4940 y Fr(k)q(M)2157 4928 y Fw(=)g Fs(b)14 b Fl(g)f Fw(.)440 5061 y(F)-7 b(or)27 b(eac)n(h)g Fs(j)h Fl(\024)22 b Fs(n)28 b Fw(w)n(e)f(then)h(ha)n(v)n(e)1194 5234 y Fs(a)1238 5246 y Fr(j)s(m)1355 5234 y Fl(\024)23 b Fs(a)1487 5246 y Fr(j)s(M)1610 5234 y Fw(+)1704 5178 y Fs(")p 1703 5215 V 1703 5291 a Fw(2)1777 5234 y Fl(\024)g Fs(a)1909 5246 y Fr(iM)2024 5234 y Fw(+)2119 5178 y Fs(")p 2117 5215 V 2117 5291 a Fw(2)2192 5234 y Fl(\024)f Fs(a)2323 5246 y Fr(im)2428 5234 y Fw(+)c Fs(")440 5413 y Fw(for)27 b(ev)n(ery)f Fs(m)d Fl(\025)g Fs(M)9 b Fw(,)27 b(so)g(that)h Fs(a)1436 5425 y Fr(j)1494 5413 y Fl(\024)23 b Fs(a)1626 5425 y Fr(i)1672 5413 y Fw(+)18 b Fs(")27 b Fw(as)g(required.)p 3244 5413 4 57 v 3248 5360 50 4 v 3248 5413 V 3297 5413 4 57 v 1851 5662 a(3)p eop %%Page: 4 4 4 3 bop 565 531 a Fw(As)37 b(for)h(Lemma)f(1)h(ab)r(o)n(v)n(e,)h(w)n(e) e(wish)h(to)f(stress)g(that)h(b)n(y)g(virtue)f(of)h(the)g(presence)440 631 y(of)e(mo)r(duli)g(the)f(algorithm)g(giv)n(en)g(through)f(the)i (pro)r(of)f(of)h(Lemma)f(2)g(alw)n(a)n(ys)f(pic)n(ks)h(a)440 731 y(de\014nite)25 b(index)g Fs(i)f Fw(ev)n(en)g(when)h(there)g(are)e (sev)n(eral)g(ones)h(ful\014lling)h(its)g(sp)r(eci\014cation.)35 b(This)440 830 y(feature)27 b(again)g(allo)n(ws)f(to)h(do)h(pro)r(ofs)e (without)i(coun)n(table)f(c)n(hoice.)565 930 y(Let)f(us)g(no)n(w)g(ha)n (v)n(e)f(a)h(lo)r(ok)f(at)h(the)h(completeness)f(prop)r(erties)f(c)n (haracteristic)f(for)i(the)440 1029 y(real)18 b(n)n(um)n(b)r(ers.)34 b(W)-7 b(e)20 b(\014rst)f(consider)f(sequen)n(tial)h(completeness.)33 b(A)20 b(sequence)f(\()p Fs(a)2973 1041 y Fr(m)3036 1029 y Fw(\))3068 1041 y Fr(m)p Fq(2)p Fp(N)3237 1029 y Fw(of)440 1129 y(reals)24 b(is)g(a)g Fv(Ca)n(uchy)j Ft(se)l(quenc)l(e)k Fw(with)25 b(mo)r(dulus)g Fs(M)18 b Fw(:)27 b Fm(Q)2186 1099 y Fu(+)2270 1129 y Fl(!)c Fm(N)35 b Fw(whenev)n(er)24 b Fl(j)p Fs(a)2890 1141 y Fr(m)2965 1129 y Fl(\000)13 b Fs(a)3087 1141 y Fr(n)3132 1129 y Fl(j)23 b(\024)g Fs(")440 1229 y Fw(for)g Fs(m;)14 b(n)23 b Fl(\025)f Fs(M)9 b Fw(\()p Fs(")p Fw(\),)25 b(and)e Ft(c)l(onver)l(ges)31 b Fw(with)24 b(mo)r(dulus)f Fs(M)18 b Fw(:)28 b Fm(Q)2325 1199 y Fu(+)2410 1229 y Fl(!)23 b Fm(N)33 b Fw(to)23 b(a)g(real)g Fs(b)p Fw(,)h(its)g Ft(limit)p Fw(,)440 1328 y(whenev)n(er)j Fl(j)p Fs(a)872 1340 y Fr(m)953 1328 y Fl(\000)18 b Fs(b)p Fl(j)23 b(\024)f Fs(")28 b Fw(for)f Fs(m)c Fl(\025)g Fs(M)9 b Fw(\()p Fs(")p Fw(\).)440 1482 y Fe(Lemma)28 b(3.)40 b Ft(Every)30 b(mo)l(dulate)l(d)38 b Fv(Ca)n(uchy)29 b Ft(se)l(quenc)l(e)g(of)g(r)l(ationals)h(c)l(onver)l (ges)g(with)f(the)440 1581 y(same)h(mo)l(dulus)g(to)f(the)h(r)l(e)l(al) g(numb)l(er)f(it)h(r)l(epr)l(esents.)440 1734 y(Pr)l(o)l(of.)43 b Fw(Let)28 b(\()p Fs(a)925 1746 y Fr(m)988 1734 y Fw(\))1020 1746 y Fr(m)p Fq(2)p Fp(N)1198 1734 y Fw(b)r(e)g(a)f Fv(Ca)n(uchy)j Fw(sequence)d(of)g(rationals)f(with)i(mo)r(dulus)g Fs(M)9 b Fw(,)28 b(and)440 1834 y(let)41 b Fs(b)f Fw(b)r(e)h(the)g (real)f(n)n(um)n(b)r(er)g(it)h(represen)n(ts.)74 b(F)-7 b(or)40 b(eac)n(h)g(rational)f Fs(")45 b(>)f Fw(0)c(and)h(ev)n(ery)440 1934 y Fs(m)29 b Fl(\025)h Fs(M)9 b Fw(\()p Fs(")p Fw(\),)32 b(w)n(e)f(ha)n(v)n(e)g(to)g(pro)n(v)n(e)f Fl(j)p Fs(a)1607 1946 y Fr(m)1691 1934 y Fl(\000)21 b Fs(b)p Fl(j)29 b(\024)g Fs(")p Fw(,)j(i.e.)g Fs(c)2227 1946 y Fr(m)2319 1934 y Fl(\025)e Fw(0)h(for)g Fs(c)2654 1946 y Fr(m)2746 1934 y Fw(=)e Fs(")21 b Fl(\000)g(j)p Fs(a)3053 1946 y Fr(m)3137 1934 y Fl(\000)g Fs(b)p Fl(j)p Fw(.)440 2033 y(No)n(w)26 b(the)g(real)g Fs(c)966 2045 y Fr(m)1055 2033 y Fw(is)g(represen)n(ted) f(b)n(y)h(the)h(sequence)2175 1966 y Fk(\000)2213 2033 y Fs(")16 b Fl(\000)f(j)p Fs(a)2415 2045 y Fr(m)2494 2033 y Fl(\000)g Fs(a)2618 2045 y Fr(k)2659 2033 y Fl(j)2682 1966 y Fk(\001)2720 2066 y Fr(k)q Fq(2)p Fp(N)2874 2033 y Fw(of)26 b(rationals,)440 2141 y(whic)n(h)k(again)f(has)h(mo)r(dulus) g Fs(M)9 b Fw(:)42 b(for)30 b(ev)n(ery)f(rational)g Fs(\021)i(>)c Fw(0)j(and)g(all)g Fs(k)s(;)14 b(l)28 b Fl(\025)f Fs(M)9 b Fw(\()p Fs(\021)s Fw(\),)32 b(w)n(e)440 2241 y(ha)n(v)n(e)485 2408 y Fl(j)p Fs(c)544 2420 y Fr(k)603 2408 y Fl(\000)18 b Fs(c)722 2420 y Fr(l)748 2408 y Fl(j)23 b Fw(=)881 2337 y Fk(\014)881 2387 y(\014)909 2408 y Fl(j)p Fs(a)976 2420 y Fr(m)1058 2408 y Fl(\000)18 b Fs(a)1185 2420 y Fr(k)1225 2408 y Fl(j)h(\000)f(j)p Fs(a)1417 2420 y Fr(m)1498 2408 y Fl(\000)g Fs(a)1625 2420 y Fr(l)1651 2408 y Fl(j)1674 2337 y Fk(\014)1674 2387 y(\014)1724 2408 y Fl(\024)1812 2337 y Fk(\014)1812 2387 y(\014)1840 2408 y Fw(\()p Fs(a)1916 2420 y Fr(m)1997 2408 y Fl(\000)g Fs(a)2124 2420 y Fr(k)2165 2408 y Fw(\))h Fl(\000)f Fw(\()p Fs(a)2375 2420 y Fr(m)2457 2408 y Fl(\000)g Fs(a)2584 2420 y Fr(l)2609 2408 y Fw(\))2641 2337 y Fk(\014)2641 2387 y(\014)2692 2408 y Fw(=)23 b Fl(j)p Fs(a)2847 2420 y Fr(k)2906 2408 y Fl(\000)18 b Fs(a)3033 2420 y Fr(l)3058 2408 y Fl(j)23 b(\024)g Fs(\021)s(:)440 2574 y Fw(Therefore)i Fs(c)851 2586 y Fr(m)937 2574 y Fl(\025)e Fw(0)j(is)g(equiv)-5 b(alen)n(t)26 b(to)g Fl(\000)p Fs(\021)g Fl(\024)d Fs(")15 b Fl(\000)h(j)p Fs(a)2090 2586 y Fr(m)2169 2574 y Fl(\000)f Fs(a)2293 2589 y Fr(M)6 b Fu(\()p Fr(\021)r Fu(\))2455 2574 y Fl(j)26 b Fw(for)g(all)g (rationals)f Fs(\021)h(>)d Fw(0.)440 2674 y(This)28 b(follo)n(ws)e (from)497 2840 y Fl(j)p Fs(a)564 2852 y Fr(m)646 2840 y Fl(\000)18 b Fs(a)773 2855 y Fr(M)6 b Fu(\()p Fr(\021)r Fu(\))934 2840 y Fl(j)24 b(\024)e(j)p Fs(a)1135 2852 y Fr(m)1217 2840 y Fl(\000)c Fs(a)1344 2855 y Fu(max)o Fq(f)p Fr(M)6 b Fu(\()p Fr(")p Fu(\))p Fr(;M)g Fu(\()p Fr(\021)r Fu(\))p Fq(g)1868 2840 y Fl(j)19 b Fw(+)f Fl(j)p Fs(a)2060 2855 y Fu(max)o Fq(f)p Fr(M)6 b Fu(\()p Fr(")p Fu(\))p Fr(;M)g Fu(\()p Fr(\021)r Fu(\))p Fq(g)2603 2840 y Fl(\000)18 b Fs(a)2730 2855 y Fr(M)6 b Fu(\()p Fr(\021)r Fu(\))2892 2840 y Fl(j)23 b(\024)g Fs(")18 b Fw(+)g Fs(\021)f(;)440 3007 y Fw(whic)n(h)28 b(holds)f(b)n(y)g(virtue)g(of)h Fs(m)23 b Fl(\025)g Fs(M)9 b Fw(\()p Fs(")p Fw(\).)p 3244 3007 4 57 v 3248 2954 50 4 v 3248 3007 V 3297 3007 4 57 v 565 3170 a(By)26 b(the)i(triangle)e(inequalit)n(y)-7 b(,)27 b(ev)n(ery)f(con)n(v)n(ergen)n(t)f(sequence)i(of)g(reals)f(with) i(mo)r(dulus)440 3269 y Fs(M)40 b Fw(is)31 b(a)g Fv(Ca)n(uchy)i Fw(sequence)e(with)h(mo)r(dulus)g Fs(")c Fl(7!)i Fs(M)9 b Fw(\()2263 3237 y Fr(")p 2262 3251 34 4 v 2262 3298 a Fu(2)2305 3269 y Fw(\).)48 b(W)-7 b(riting)31 b Fl(d)p Fs(")p Fl(e)g Fw(for)g(the)h(least)440 3369 y(in)n(teger)27 b Fl(\025)22 b Fs(")28 b Fw(whenev)n(er)e Fs(")d(>)g Fw(0)k(is)g(rational,)g(w)n(e)g(no)n(w)g(pro)n(v)n(e)f(the)i(rev)n (erse)d(implication.)3236 3339 y Fu(4)440 3522 y Fe(Theorem)j(4)g (\(Sequen)m(tial)h(Completeness\).)36 b Ft(F)-6 b(or)28 b(every)g(mo)l(dulate)l(d)37 b Fv(Ca)n(uchy)27 b Ft(se-)440 3622 y(quenc)l(e)i(of)i(r)l(e)l(als)f(we)g(c)l(an)g(\014nd)f(a)i(r)l(e) l(al)f(numb)l(er)f(to)g(which)j(it)d(c)l(onver)l(ges)i(with)f(mo)l (dulus.)440 3775 y(Pr)l(o)l(of.)43 b Fw(Let)37 b(\()p Fs(a)934 3787 y Fr(m)998 3775 y Fw(\))1030 3787 y Fr(m)p Fq(2)p Fp(N)1217 3775 y Fw(b)r(e)g(the)g Fv(Ca)n(uchy)i Fw(sequence)e(of)g(reals)f(with)h(mo)r(dulus)g Fs(M)9 b Fw(;)42 b(for)440 3875 y(ev)n(ery)18 b Fs(m)23 b Fl(2)h Fm(N)t Fw(,)j(let)20 b Fs(a)1086 3887 y Fr(m)1168 3875 y Fw(b)r(e)g(represen)n(ted)e(b)n(y)i(a)f Fv(Ca)n(uchy)i Fw(sequence)e(\()p Fs(a)2624 3887 y Fr(mk)2724 3875 y Fw(\))2756 3887 y Fr(k)q Fq(2)p Fp(N)2903 3875 y Fw(of)h(rationals)440 3974 y(with)32 b(mo)r(dulus)f Fs(M)1052 3986 y Fr(m)1115 3974 y Fw(.)48 b(Note)31 b(\014rst)g(that,)i(for)d(eac)n(h)h Fs(m)e Fl(2)g Fm(N)42 b Fw(and)31 b(ev)n(ery)f(rational)g Fs(\021)i(>)d Fw(0,)440 4074 y(b)n(y)e(Lemma)h(3)f(w)n(e)g(ha)n(v)n(e)f Fl(j)p Fs(a)1301 4086 y Fr(m)1383 4074 y Fl(\000)18 b Fs(a)1510 4086 y Fr(ml)1594 4074 y Fl(j)23 b(\024)1751 4041 y Fu(1)p 1738 4055 59 4 v 1738 4102 a Fr(m)1835 4074 y Fw(for)k(all)g Fs(l)d Fl(\025)f Fs(M)2295 4086 y Fr(m)2358 4074 y Fw(\()2413 4041 y Fu(1)p 2400 4055 V 2400 4102 a Fr(m)2469 4074 y Fw(\).)37 b(Next,)28 b(set)1581 4240 y Fs(b)1617 4252 y Fr(m)1703 4240 y Fw(=)23 b Fs(a)1835 4282 y Fr(mM)1957 4290 y Fj(m)2012 4215 y Fk(\000)2071 4260 y Fi(1)p 2060 4269 51 3 v 2060 4302 a Fj(m)2121 4215 y Fk(\001)440 4457 y Fw(for)31 b(ev)n(ery)f Fs(m)f Fl(2)g Fm(N)t Fw(,)39 b(so)30 b(that)i Fl(j)p Fs(a)1452 4469 y Fr(m)1536 4457 y Fl(\000)20 b Fs(b)1657 4469 y Fr(m)1720 4457 y Fl(j)29 b(\024)1889 4424 y Fu(1)p 1876 4438 59 4 v 1876 4485 a Fr(m)1976 4457 y Fw(for)i(all)g Fs(m)e Fl(2)g Fm(N)42 b Fw(b)n(y)31 b(the)g(particular)f(case)440 4556 y Fs(l)24 b Fw(=)f Fs(M)658 4568 y Fr(m)721 4489 y Fk(\000)782 4524 y Fu(1)p 769 4538 V 769 4585 a Fr(m)838 4489 y Fk(\001)903 4556 y Fw(of)28 b(the)g(foregoing)e(consideration.) 35 b(Then)701 4770 y Fl(j)p Fs(b)760 4782 y Fr(m)841 4770 y Fl(\000)18 b Fs(b)960 4782 y Fr(n)1005 4770 y Fl(j)23 b(\024)g(j)p Fs(b)1198 4782 y Fr(m)1279 4770 y Fl(\000)18 b Fs(a)1406 4782 y Fr(m)1469 4770 y Fl(j)h Fw(+)f Fl(j)p Fs(a)1661 4782 y Fr(m)1742 4770 y Fl(\000)g Fs(a)1869 4782 y Fr(n)1914 4770 y Fl(j)h Fw(+)f Fl(j)p Fs(a)2106 4782 y Fr(n)2169 4770 y Fl(\000)g Fs(b)2288 4782 y Fr(n)2333 4770 y Fl(j)23 b(\024)2493 4714 y Fw(1)p 2477 4751 73 4 v 2477 4827 a Fs(m)2578 4770 y Fw(+)2673 4714 y Fs(")p 2671 4751 42 4 v 2671 4827 a Fw(2)2741 4770 y(+)2838 4714 y(1)p 2834 4751 50 4 v 2834 4827 a Fs(n)2917 4770 y Fl(\024)g Fs(")440 4973 y Fw(for)28 b(all)g Fs(m;)14 b(n)24 b Fl(\025)g Fs(N)9 b Fw(\()p Fs(")p Fw(\))25 b(=)f(max)1418 4906 y Fk(\010)1467 4973 y Fs(M)9 b Fw(\()1600 4940 y Fr(")p 1599 4954 34 4 v 1599 5002 a Fu(2)1642 4973 y Fw(\))p Fs(;)14 b Fl(d)1758 4940 y Fu(4)p 1758 4954 V 1759 5002 a Fr(")1801 4973 y Fl(e)1838 4906 y Fk(\011)1886 4973 y Fw(,)29 b(whic)n(h)f(is)h(to)f (sa)n(y)f(that)i(\()p Fs(b)2756 4985 y Fr(m)2819 4973 y Fw(\))g(is)f(a)g(Cauc)n(h)n(y)440 5073 y(sequence)d(with)h(mo)r (dulus)g Fs(N)9 b Fw(,)26 b(and)f(therefore)g(represen)n(ts)f(a)h(real) g(n)n(um)n(b)r(er)g Fs(b)p Fw(.)36 b(Moreo)n(v)n(er,)1045 5274 y Fl(j)p Fs(a)1112 5286 y Fr(m)1194 5274 y Fl(\000)18 b Fs(b)p Fl(j)23 b(\024)f(j)p Fs(a)1513 5286 y Fr(m)1595 5274 y Fl(\000)c Fs(b)1714 5286 y Fr(m)1777 5274 y Fl(j)g Fw(+)g Fl(j)p Fs(b)1960 5286 y Fr(m)2041 5274 y Fl(\000)g Fs(b)p Fl(j)23 b(\024)2319 5218 y Fw(1)p 2304 5255 73 4 v 2304 5331 a Fs(m)2405 5274 y Fw(+)2499 5218 y Fs(")p 2498 5255 42 4 v 2498 5331 a Fw(2)2572 5274 y Fl(\024)g Fs(")p 440 5336 1146 4 v 532 5390 a Fo(4)567 5413 y Fn(This)g(stems)g (from)f(lecture)i(notes)h(of)e(the)i(second)f(author.)1851 5662 y Fw(4)p eop %%Page: 5 5 5 4 bop 440 531 a Fw(for)23 b(all)h Fs(m)f Fl(\025)f Fs(N)9 b Fw(\()977 499 y Fr(")p 976 513 34 4 v 976 560 a Fu(2)1020 531 y Fw(\))23 b(=)g(max)1331 464 y Fk(\010)1379 531 y Fs(M)9 b Fw(\()1512 499 y Fr(")p 1511 513 V 1511 560 a Fu(4)1554 531 y Fw(\))p Fs(;)14 b Fl(d)1670 499 y Fu(8)p 1670 513 V 1671 560 a Fr(")1713 531 y Fl(e)1750 464 y Fk(\011)1799 531 y Fw(;)25 b(in)f(other)f(w)n(ords:)34 b(\()p Fs(a)2498 543 y Fr(m)2561 531 y Fw(\))24 b(con)n(v)n(erges)e(to) h Fs(b)h Fw(with)440 631 y(mo)r(dulus)k Fs(")23 b Fl(7!)g Fs(N)9 b Fw(\()1062 598 y Fr(")p 1061 612 V 1061 660 a Fu(2)1104 631 y Fw(\).)p 3244 631 4 57 v 3248 578 50 4 v 3248 631 V 3297 631 4 57 v 565 793 a(This)25 b(pro)r(of)h(happ)r (ens)g(not)f(to)h(need)g(coun)n(table)f(c)n(hoice)g(but)i(requires)d(a) i(certain)f(con-)440 892 y(ceptual)j(c)n(hoice)f(made)h(in)g(adv)-5 b(ance:)36 b(that)29 b(eac)n(h)e(real)g(n)n(um)n(b)r(er)g(is)h(mean)n (t)g(to)f(b)r(e)i(nothing)440 992 y(but)36 b(a)f(mo)r(dulated)g Fv(Ca)n(uchy)i Fw(sequences)e(of)g(rationals.)58 b(Giv)n(en)35 b(this)h(presupp)r(osition,)440 1091 y(it)30 b(seems)e(to)h(cause)g(no) g(reserv)-5 b(ations)27 b(from)i(the)g(c)n(hoice-free)f(p)r(ersp)r (ectiv)n(e)h(that)g(a)g(single)440 1191 y(sequence)e(of)h(reals)e(is)i (understo)r(o)r(d)f(as)g(a)g(double)g(sequence)h(of)f(rationals.)2807 1161 y Fu(5)565 1291 y Fw(On)36 b(the)i(other)e(hand,)j(coun)n(table)d (c)n(hoice)g(w)n(ould)h(b)r(ecome)g(indisp)r(ensable)g(for)f(the)440 1390 y(pro)r(of)27 b(of)g(sequen)n(tial)g(completeness)g(as)g(so)r(on)f (as)h(the)h(concept)f(of)h(reals)e(w)n(as)g(relaxed)h(ei-)440 1490 y(ther)22 b(to)g Fv(Ca)n(uchy)i Fw(sequences)d(of)h(rationals)f (without)h(mo)r(duli)h(or)e(to)h(equiv)-5 b(alence)22 b(classes,)440 1590 y(mo)r(dulo)28 b(=,)f(of)g(mo)r(dulated)h Fv(Ca)n(uchy)i Fw(sequences)c(of)i(rationals.)35 b(T)-7 b(o)28 b(start)f(our)f(pro)r(of)h(of)440 1689 y(Theorem)35 b(4)g(in)h(the)f(former)g(case,)i(one)e(had)g(to)g(c)n(ho)r(ose,)i(for) e(ev)n(ery)f Fs(m)p Fw(,)j(the)f(in)n(teger)f Fs(l)440 1789 y Fw(with)29 b Fl(j)p Fs(a)697 1801 y Fr(m)780 1789 y Fl(\000)19 b Fs(a)908 1801 y Fr(ml)992 1789 y Fl(j)25 b(\024)1153 1756 y Fu(1)p 1140 1770 59 4 v 1140 1817 a Fr(m)1209 1789 y Fw(,)k(whic)n(h)g(in)g(the)g(presence)f(of)h(the)g (mo)r(dulus)g Fs(M)2733 1801 y Fr(m)2825 1789 y Fw(of)f Fs(a)2964 1801 y Fr(m)3056 1789 y Fw(w)n(e)h(can)440 1888 y(simply)c(de\014ne)h(b)n(y)e(setting)h Fs(l)g Fw(=)e Fs(M)1542 1900 y Fr(m)1604 1888 y Fw(\()1659 1856 y Fu(1)p 1646 1870 V 1646 1917 a Fr(m)1715 1888 y Fw(\).)37 b(In)25 b(the)g(latter)g(case,)g(one)g(had)g(to)g(c)n(ho)r(ose)f(\014rst,)440 1988 y(again)k(for)h(ev)n(ery)f Fs(m)p Fw(,)i(a)f(represen)n(tativ)n(e) f(\()p Fs(a)1822 2000 y Fr(mk)1922 1988 y Fw(\))1954 2000 y Fr(k)q Fq(2)p Fp(N)2111 1988 y Fw(of)h(the)h(real)e Fs(a)2559 2000 y Fr(m)2622 1988 y Fw(,)i(whic)n(h)g(w)n(e)f(already)440 2088 y(understand)e(as)g(to)h(consist)f(in)h(nothing)f(but)h(suc)n(h)f (a)h(sequence)f(of)g(rationals.)565 2187 y(F)-7 b(ollo)n(wing)21 b([2)o(,)h(3,)g(5])g(in)g(the)h(st)n(yle)e(of)i([18)o(],)g(let)f(us)h (no)n(w)e(consider)g(order)g(completeness.)440 2287 y(T)-7 b(o)27 b(this)h(end,)g(let)g Fs(S)k Fw(b)r(e)c(a)f(set)g(of)h(real)f(n) n(um)n(b)r(ers.)36 b(A)28 b(real)e Fs(b)i Fw(is)f(an)g Ft(upp)l(er)j(b)l(ound)36 b Fw(of)28 b Fs(S)5 b Fw(,)27 b(or)440 2387 y Fs(S)36 b Fw(is)30 b Ft(b)l(ounde)l(d)j(fr)l(om)g(ab)l (ove)39 b Fw(b)n(y)30 b Fs(b)p Fw(,)i(whenev)n(er)d Fs(s)g Fl(\024)f Fs(b)i Fw(for)g(all)h Fs(s)d Fl(2)h Fs(S)5 b Fw(,)31 b(and)g(a)f Ft(le)l(ast)j(upp)l(er)440 2486 y(b)l(ound)j Fw(or)26 b Ft(supr)l(emum)35 b Fw(sup)14 b Fs(S)31 b Fw(of)c Fs(S)32 b Fw(if,)c(in)f(addition,)g(for)g(ev)n(ery) f(real)g Fs(a)h Fw(with)g Fs(a)c(<)g(b)k Fw(there)440 2586 y(is)g(an)h Fs(r)e Fl(2)d Fs(S)32 b Fw(with)d Fs(r)c(>)e(a)p Fw(.)1270 2556 y Fu(6)565 2685 y Fw(If)j Fs(b)g Fw(is)g(a)g(least)g (upp)r(er)g(b)r(ound)h(and)f Fs(a)g Fw(an)g(upp)r(er)g(b)r(ound)h(of)f Fs(S)5 b Fw(,)26 b(then)h Fs(a)c(<)g(b)j Fw(is)g(imp)r(os-)440 2785 y(sible,)g(so)f(that)h Fs(b)c Fl(\024)h Fs(a)p Fw(;)j(whence)g(up) g(to)f(equalit)n(y)g(there)g(is)h(at)f(most)h(one)f(suprem)n(um)g(of)h Fs(S)5 b Fw(.)440 2885 y(If)32 b Fs(S)37 b Fw(has)31 b(a)g(suprem)n(um,)i(then)f Fs(S)k Fw(is)c(inhabited)g(and)g(b)r (ounded)g(from)f(ab)r(o)n(v)n(e,)h(but)g(ev)n(en)440 2984 y(when)d Fs(S)k Fw(has)28 b(the)h(t)n(w)n(o)f(latter)g(prop)r (erties,)g(the)h(former)f(is)g(not)h(gran)n(ted)e(constructiv)n(ely)-7 b(.)440 3084 y(A)26 b(simple)g(exception)f(is)h(the)g(maxim)n(um)f(of)h (\014nitely)g(man)n(y)f(reals)f(\(see)i(ab)r(o)n(v)n(e\),)f(whic)n(h)g (of)440 3184 y(course)h(is)i(the)g(suprem)n(um)f(thereof.)565 3283 y(A)39 b(more)f(relev)-5 b(an)n(t)39 b(su\016cien)n(t)g(condition) g(for)f(the)i(constructiv)n(e)e(existence)h(of)g(the)440 3383 y(suprem)n(um)d(of)h(a)f(general)f(set)i Fs(S)k Fw(of)c(real)e(n)n(um)n(b)r(ers)i(\(b)r(eing)g(inhabited)g(and)f(b)r (ounded)440 3482 y(from)27 b(ab)r(o)n(v)n(e\))g(it)h(that)g Fs(S)k Fw(b)r(e)c Ft(or)l(der)j(lo)l(c)l(ate)l(d)f(fr)l(om)g(ab)l(ove) 2247 3452 y Fu(7)2286 3482 y Fw(,)e(whic)n(h)f(is)h(to)f(sa)n(y)g(that) 730 3643 y(\006)790 3655 y Fr(S)838 3643 y Fw(\()p Fs(a;)14 b(b)p Fw(\):)203 b(either)27 b Fs(s)c Fl(\024)g Fs(b)k Fw(for)g(all)g Fs(s)c Fl(2)h Fs(S)32 b Fw(or)27 b Fs(a)c(<)f(r)31 b Fw(for)c(some)g Fs(r)f Fl(2)d Fs(S)440 3804 y Fw(holds)29 b(for)f(all)h(pairs)g Fs(a;)14 b(b)28 b Fw(of)h(real)g(n)n(um)n(b)r (ers)f(with)i Fs(a)25 b(<)h(b)p Fw(.)41 b(T)-7 b(o)29 b(aim)g(at)g(the)g(su\016ciency)g(of)440 3903 y(these)f(three)f (conditions,)g(it)h(is)g(useful)g(to)f(consider)718 4064 y(\005)780 4076 y Fr(S)829 4064 y Fw(\()p Fs(a;)14 b(b)p Fw(\):)203 b(b)r(oth)28 b Fs(s)23 b Fl(\024)f Fs(b)27 b Fw(for)h(all)f Fs(s)c Fl(2)g Fs(S)33 b Fw(and)27 b Fs(a)c(<)g(r)30 b Fw(for)d(some)g Fs(r)f Fl(2)d Fs(S)440 4224 y Fw(as)31 b(a)h(prop)r(ert)n(y)f(of)h(an)n(y)g(pair)f Fs(a;)14 b(b)32 b Fw(of)g(real)f(n)n(um)n(b)r(ers)g(with)i Fs(a)d(<)h(b)p Fw(.)50 b(Note)32 b(that)g(\006)3075 4236 y Fr(S)3123 4224 y Fw(\()p Fs(a;)14 b(b)p Fw(\))440 4324 y(and)26 b(\005)662 4336 y Fr(S)710 4324 y Fw(\()p Fs(a;)14 b(b)p Fw(\))26 b(are)f(the)i(disjunction)f(and)g(conjunction,)h(resp)r (ectiv)n(ely)-7 b(,)25 b(of)h(the)h(same)e(pair)440 4423 y(of)j(prop)r(ositions.)p 440 4468 1146 4 v 532 4521 a Fo(5)567 4545 y Fn(When)c(one)f(w)n(an)n(ts)h(to)f(read)g(this)f(pro) r(of)h(as)f(a)h(completeness)h(pro)r(of)e(for)g(the)h(sequen)n(tial)h (completion)440 4624 y(of)h(an)g(arbitrary)f(metric)g(space,)i(one)f (again)g(has)g(to)h(view)f(sequences)h(in)f(this)g(completion)g({)f (the)i(space)440 4703 y(of)h(\(mo)r(dulated\))h Fd(Ca)o(uchy)f Fn(sequences)i(in)e(the)g(original)g(space)h({)f(as)g(double)h (sequences)g(in)f(the)h(latter.)440 4781 y(An)n(y)g(suc)n(h)h(in)n (terpretation)h(ma)n(y)d(only)h(lo)r(ok)h(somewhat)f(strange)h(in)f (cases)h(whic)n(h)f(are)g(more)f(complex)440 4860 y(than)e(that)g(of)e (the)h(completion)g(of)g(the)g(rationals)g(to)g(the)g(reals.)511 4939 y(F)-6 b(or)28 b(this)g(observ)l(ation,)i(and)e(for)g(man)n(y)f (others)i(on)g(the)g(role)e(of)h(coun)n(table)i(c)n(hoice)g(in)d (constructiv)n(e)440 5018 y(mathematics,)j(the)g(\014rst)f(author)h(is) f(indebted)i(to)f(F)-6 b(red)30 b Fd(Richman)p Fn(,)f(for)g(whose)h(p)r (osition)f(w)n(e)h(refer)f(to)440 5097 y([15].)532 5153 y Fo(6)567 5176 y Fn(This)c(is)f(a)i(p)r(ositiv)n(e)f(w)n(a)n(y)h(to)g (sa)n(y)f(that)i(ev)n(ery)f(real)f Fc(a)g Fn(with)h Fc(a)c(<)g(b)p Fn(,)k(ho)n(w)n(ev)n(er)g(close)f(it)h(ma)n(y)e(b)r(e)i(to)f Fc(b)p Fn(,)440 5255 y(is)e(no)h(upp)r(er)g(b)r(ound)h(of)e Fc(S)t Fn(.)532 5311 y Fo(7)567 5334 y Fn(This)c(name)h(w)n(as)g (coined)g(in)g([8].)29 b(The)20 b(same)f(prop)r(ert)n(y)i(has)f(b)r (een)g(called)g(\\lo)r(cated)i(ab)r(o)n(v)n(e")f(b)n(y)g(P)n(eter)440 5413 y Fd(A)n(czel)j Fn(and)h(Mic)n(hael)e Fd(Ra)l(thjen)g Fn([1],)g(and)h(\\lo)n(w)n(er)g(lo)r(cated")h(b)n(y)f(Erik)f Fd(P)-6 b(almgren)23 b Fn([11].)1851 5662 y Fw(5)p eop %%Page: 6 6 6 5 bop 440 531 a Fe(Theorem)32 b(5)i(\(Order)g(Completeness\).)39 b Ft(L)l(et)31 b Fs(S)g Fl(\032)25 b Fm(R)38 b Ft(b)l(e)31 b(inhabite)l(d,)j(b)l(ounde)l(d)e(fr)l(om)440 631 y(ab)l(ove,)f(and)g (or)l(der)f(lo)l(c)l(ate)l(d)h(fr)l(om)f(ab)l(ove.)40 b(Then)30 b Fs(S)35 b Ft(has)30 b(a)g(le)l(ast)g(upp)l(er)g(b)l(ound.) 440 787 y(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)34 b(ma)n(y)f(assume)f(that)i (w)n(e)f(ha)n(v)n(e)f Fs(a;)14 b(b)32 b Fl(2)h Fm(Q)45 b Fw(with)34 b Fs(a)f(<)f(b)h Fw(suc)n(h)g(that)h(\005)3052 799 y Fr(S)3100 787 y Fw(\()p Fs(a;)14 b(b)p Fw(\).)440 886 y(W)-7 b(e)28 b(construct)f Fs(c;)14 b(d)9 b Fw(:)28 b Fm(N)33 b Fl(!)23 b Fm(Q)39 b Fw(suc)n(h)28 b(that)g(for)f(all)g Fs(m)970 1056 y(a)c Fw(=)f Fs(c)1160 1068 y Fu(0)1220 1056 y Fl(\024)h Fs(c)1344 1068 y Fu(1)1404 1056 y Fl(\024)g(\001)14 b(\001)g(\001)23 b(\024)f Fs(c)1735 1068 y Fr(m)1822 1056 y Fs(<)g(d)1952 1068 y Fr(m)2039 1056 y Fl(\024)g(\001)14 b(\001)g(\001)23 b(\024)g Fs(d)2377 1068 y Fu(1)2437 1056 y Fl(\024)g Fs(d)2568 1068 y Fu(0)2628 1056 y Fw(=)g Fs(b;)423 b Fw(\(1\))970 1181 y(\005)1032 1193 y Fr(S)1080 1181 y Fw(\()p Fs(c)1148 1193 y Fr(m)1211 1181 y Fs(;)14 b(d)1291 1193 y Fr(m)1354 1181 y Fw(\))p Fs(;)1789 b Fw(\(2\))970 1354 y Fs(d)1013 1366 y Fr(m)1094 1354 y Fl(\000)18 b Fs(c)1213 1366 y Fr(m)1299 1354 y Fl(\024)1387 1262 y Fk(\020)1447 1298 y Fw(2)p 1447 1335 42 4 v 1447 1411 a(3)1498 1262 y Fk(\021)1548 1279 y Fr(m)1611 1354 y Fw(\()p Fs(b)g Fl(\000)g Fs(a)p Fw(\))p Fs(:)1319 b Fw(\(3\))440 1552 y(Let)31 b Fs(c)628 1564 y Fu(0)665 1552 y Fs(;)14 b(:)g(:)g(:)g(;)g(c)886 1564 y Fr(m)980 1552 y Fw(and)31 b Fs(d)1188 1564 y Fu(0)1225 1552 y Fs(;)14 b(:)g(:)g(:)g(;)g(d)1453 1564 y Fr(m)1547 1552 y Fw(b)r(e)32 b(already)d(constructed)i(suc)n(h)g(that)g(\(3\))g (holds.)47 b(Let)440 1651 y Fs(x)24 b Fw(=)e Fs(c)634 1663 y Fr(m)715 1651 y Fw(+)808 1619 y Fu(1)p 808 1633 34 4 v 808 1680 a(3)851 1651 y Fw(\()p Fs(d)926 1663 y Fr(m)1007 1651 y Fl(\000)c Fs(c)1126 1663 y Fr(m)1189 1651 y Fw(\))28 b(and)f Fs(y)f Fw(=)c Fs(c)1600 1663 y Fr(m)1681 1651 y Fw(+)1774 1619 y Fu(2)p 1774 1633 V 1774 1680 a(3)1817 1651 y Fw(\()p Fs(d)1892 1663 y Fr(m)1974 1651 y Fl(\000)17 b Fs(c)2092 1663 y Fr(m)2155 1651 y Fw(\).)38 b(Since)27 b Fs(S)32 b Fw(is)c(order)e(lo)r(cated)h (from)440 1751 y(ab)r(o)n(v)n(e,)35 b(either)g Fs(s)g Fl(\024)g Fs(y)j Fw(for)c(all)h Fs(s)g Fl(2)h Fs(S)j Fw(or)34 b(else)h Fs(x)h(<)f(r)i Fw(for)e(some)f Fs(r)k Fl(2)e Fs(S)5 b Fw(.)58 b(In)35 b(the)h(\014rst)440 1850 y(case)29 b(let)i Fs(c)777 1862 y Fr(m)p Fu(+1)951 1850 y Fw(:=)c Fs(c)1102 1862 y Fr(m)1195 1850 y Fw(and)j Fs(d)1402 1862 y Fr(m)p Fu(+1)1576 1850 y Fw(:=)d Fs(y)s Fw(,)j(and)g(in)h(the)f(second)f(case)h(let)g Fs(c)2804 1862 y Fr(m)p Fu(+1)2978 1850 y Fw(:=)d Fs(x)k Fw(and)440 1950 y Fs(d)483 1962 y Fr(m)p Fu(+1)663 1950 y Fw(:=)i Fs(d)827 1962 y Fr(m)890 1950 y Fw(.)55 b(Then)34 b(clearly)e(\005)1524 1962 y Fr(S)1573 1950 y Fw(\()p Fs(c)1641 1962 y Fr(m)p Fu(+1)1788 1950 y Fs(;)14 b(d)1868 1962 y Fr(m)p Fu(+1)2015 1950 y Fw(\),)36 b(and)d(also)g(\(1\))g(and)h(\(3\))f(con)n(tin)n(ue)g (to)440 2050 y(hold)28 b(for)g Fs(m)19 b Fw(+)g(1.)38 b(W)-7 b(e)29 b(claim)f(that)h(the)g(real)e(n)n(um)n(b)r(ers)h Fs(c)c Fw(=)g Fs(d)29 b Fw(giv)n(en)e(b)n(y)h(the)h(mo)r(dulated)440 2149 y Fv(Ca)n(uchy)37 b Fw(sequence)e(of)g(rationals)f(\()p Fs(c)1665 2161 y Fr(m)1728 2149 y Fw(\))i(and)f(\()p Fs(d)2040 2161 y Fr(m)2103 2149 y Fw(\))h(is)f(the)h(least)e(upp)r(er)i (b)r(ound)f(of)h Fs(S)5 b Fw(.)440 2249 y(Indeed,)25 b(it)f(is)g(an)g(upp)r(er)g(b)r(ound)h(b)r(ecause)e(for)h Fs(")f(>)f Fw(0)i(w)n(e)g(ha)n(v)n(e)e Fs(d)i Fl(\024)e Fs(d)2633 2261 y Fr(m)2720 2249 y Fs(<)g(d)11 b Fw(+)g Fs(")24 b Fw(for)g(some)440 2349 y Fs(m)h Fw(and)f Fs(d)739 2361 y Fr(m)827 2349 y Fw(is)h(an)f(upp)r(er)h(b)r(ound)g(of)f Fs(S)5 b Fw(,)25 b(and)g(similarly)f Fs(c)13 b Fl(\000)g Fs(")21 b(<)h(c)2509 2361 y Fr(m)2596 2349 y Fl(\024)g Fs(c)j Fw(for)f(some)g Fs(m)h Fw(and)440 2448 y Fs(c)476 2460 y Fr(m)562 2448 y Fs(<)e(r)30 b Fw(for)d(some)g Fs(r)f Fl(2)d Fs(S)5 b Fw(.)p 3244 2448 4 57 v 3248 2396 50 4 v 3248 2448 V 3297 2448 4 57 v 565 2612 a(One)23 b(migh)n(t)g(think)h(that)f(dep)r(enden)n(t)h(c)n(hoice)f(has)g(b)r (een)h(used)f(this)h(pro)r(of:)34 b(in)23 b(the)h(case)440 2711 y(distiction)g(\\either)f Fs(s)g Fl(\024)g Fs(y)j Fw(for)d(all)h Fs(s)f Fl(2)g Fs(S)29 b Fw(or)23 b(else)g Fs(x)h(<)e(r)27 b Fw(for)c(some)g Fs(r)j Fl(2)d Fs(S)5 b Fw(")23 b(the)h(cases)f(ma)n(y)440 2811 y(w)n(ell)30 b(o)n(v)n(erlap.)43 b(Ho)n(w)n(ev)n(er,)30 b(reference)f(to)h(the)h (algorithm)f(pro)n(vided)f(b)n(y)h(the)h(assumption)440 2911 y(that)d Fs(S)k Fw(is)c(order)e(lo)r(cated)h(from)h(ab)r(o)n(v)n (e)e(mak)n(e)h(this)g(c)n(hoice)g(sup)r(er\015uous.)565 3010 y(Another)33 b(imp)r(ortan)n(t)h(condition)f(su\016cien)n(t)h(for) f(the)h(existence)g(of)g(a)f(suprem)n(um)g(is)440 3110 y(that)28 b Fs(S)k Fw(is)c(a)f Ft(total)t(ly)k(b)l(ounde)l(d)36 b Fw(set)28 b(of)f(reals.)36 b(Let)28 b(us)f(consider)g(this)h(concept) f(next.)565 3210 y(T)-7 b(o)31 b(start)g(with,)i(let)f(us)f(call)g(an)g (arbitrary)f(set)h Ft(\014nite)38 b Fw(if)32 b(there)f(is)g(a)g (mapping)h(from)440 3309 y Fl(f)p Fw(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(n)p Fl(g)28 b Fw(on)n(to)g(that)h(set)g(with)g Fs(n)c Fl(\025)g Fw(1;)k(in)g(particular,)f(\014nite)h(sets)g(are)f(inhabited.)3164 3279 y Fu(8)3242 3309 y Fw(A)440 3409 y(set)36 b Fs(S)k Fw(of)35 b(real)g(n)n(um)n(b)r(ers)g(is)h Ft(total)t(ly)i(b)l(ounde)l (d)44 b Fw(whenev)n(er)35 b(for)g(ev)n(ery)f(\(rational\))h Fs(")h(>)g Fw(0)440 3508 y(there)28 b(is)f(an)h Fs(")p Ft(-appr)l(oximation)35 b Fw(to)28 b Fs(S)5 b Fw(:)37 b(that)28 b(is,)f(a)h(\014nite)g(subset)g Fs(T)39 b Fw(of)27 b Fs(S)33 b Fw(suc)n(h)27 b(that)h(eac)n(h)440 3608 y(elemen)n(t)21 b(of)f Fs(S)25 b Fw(is)20 b(within)h Fs(")g Fw(of)f(an)g(elemen)n(t)g (of)h Fs(T)12 b Fw(.)33 b(T)-7 b(otally)20 b(b)r(ounded)h(sets)f(of)g (real)g(n)n(um)n(b)r(ers)440 3708 y(are)27 b(ob)n(viously)f(inhabited)i (and)f(b)r(ounded.)565 3807 y(Note)35 b(that)h(the)g(de\014nition)g(of) g(total)f(b)r(oundedness)h(in)n(v)n(olv)n(es)e(t)n(w)n(o)h(witnessing)g (ob-)440 3907 y(jects:)52 b(the)35 b(sequence)f(\()p Fs(T)1261 3919 y Fr(m)1324 3907 y Fw(\))1356 3919 y Fr(m)p Fq(\025)p Fu(1)1539 3907 y Fw(of)h(\014nite)h(subsets)e(of)h Fs(S)40 b Fw(so)34 b(that)h(eac)n(h)f Fs(T)2887 3919 y Fr(m)2984 3907 y Fw(is)h(an)3221 3874 y Fu(1)p 3208 3888 59 4 v 3208 3936 a Fr(m)3277 3907 y Fw(-)440 4007 y(appro)n(ximation)j(to)i Fs(S)5 b Fw(,)43 b(and)d(also)e(a)i(function) h(assigning)d(to)i(eac)n(h)f(elemen)n(t)h(of)g Fs(S)45 b Fw(an)440 4106 y Fs(")p Fw(-close)25 b(elemen)n(t)h(of)g Fs(T)12 b Fw(.)36 b(Notice)26 b(also)f(that)i(the)f(latter)g(function,) h(ha)n(ving)e(a)h(\014nite)h(range,)440 4206 y(is)g(necessarily)f (non-con)n(tin)n(uous.)565 4305 y(Clearly)c(ev)n(ery)g(totally)i(b)r (ounded)g(set)f(of)h(reals)e(is)i(order)e(lo)r(cated)h(from)g(ab)r(o)n (v)n(e;)h(there-)440 4405 y(fore)j(w)n(e)g(immediately)h(obtain)f(from) h(Theorem)e(5:)440 4561 y Fe(Corollary)33 b(6.)41 b Ft(Every)31 b(total)t(ly)f(b)l(ounde)l(d)g(set)g(of)g(r)l(e)l(al)g(numb)l(ers)f (has)i(a)f(supr)l(emum.)565 4717 y Fw(As)d(a)g(digression,)f(let)i(us)g (recall)f(the)h(standard)e(application)h(of)h(Corollary)d(6.)440 4873 y Fe(Prop)s(osition)35 b(7.)44 b Ft(L)l(et)33 b Fs(D)f Fl(\032)e Fm(R)40 b Ft(and)34 b Fs(f)18 b Fw(:)29 b Fs(D)j Fl(!)e Fm(R)40 b Ft(a)34 b(uniformly)h(c)l(ontinuous)e (function.)440 4972 y(If)d(the)g(domain)h Fs(D)h Ft(of)e Fs(f)39 b Ft(is)30 b(total)t(ly)h(b)l(ounde)l(d,)f(then)g(so)g(is)g (the)g(r)l(ange)g Fs(f)9 b Fw(\()p Fs(D)r Fw(\))30 b Ft(of)g Fs(f)9 b Ft(.)p 440 5020 1146 4 v 532 5074 a Fo(8)567 5098 y Fn(In)19 b(this)e(resp)r(ect)i(w)n(e)g(follo)n(w)e([2,) g(3])h(rather)g(than)h([5,)e(10)q(],)h(where)g(\014nite)h(sets)f(are)g (allo)n(w)n(ed)g(to)h(b)r(e)f(empt)n(y)-6 b(.)440 5176 y(F)g(or)22 b(the)g(sak)n(e)h(of)e(simplicit)n(y)-6 b(,)21 b(w)n(e)h(also)f(use)h(\\\014nite")i(instead)e(of)g(the)h(constructiv)n (ely)g(more)e(correct)h(term)440 5255 y(\\\014nitely)j(en)n(umerable")g (\(due)g(to)g(Ra)n(y)h Fd(Mines)p Fn(\))e(or)h(\\sub\014nite")g (\(coined)h(b)n(y)f(Errett)f Fd(Bishop)p Fn(\).)35 b(Usually)440 5334 y(\\\014nite")f(is)e(reserv)n(ed)h(for)f(the)i(sets)f(that)h(are)f (in)f(a)h(bijectiv)n(e)h(corresp)r(ondence)g(to)g(sets)f(of)f(the)i (form)440 5413 y Fg(f)p Fn(1)p Fc(;)12 b(:)f(:)h(:)f(;)g(n)p Fg(g)p Fn(,)24 b(and)g(whic)n(h)g(necessarily)f(are)h(discrete.)1851 5662 y Fw(6)p eop %%Page: 7 7 7 6 bop 440 531 a Ft(Pr)l(o)l(of.)43 b Fw(F)-7 b(or)36 b Fs(")i(>)g Fw(0,)g(pic)n(k)f Fs(\016)k(>)d Fw(0)e(so)g(that,)j(for)e (all)f Fs(x;)14 b(y)41 b Fl(2)e Fs(D)r Fw(,)g(if)e Fl(j)p Fs(x)25 b Fl(\000)f Fs(y)s Fl(j)38 b(\024)g Fs(\016)s Fw(,)h(then)440 631 y Fl(j)p Fs(f)9 b Fw(\()p Fs(x)p Fw(\))23 b Fl(\000)e Fs(f)9 b Fw(\()p Fs(y)s Fw(\))p Fl(j)33 b(\024)e Fs(")p Fw(.)54 b(No)n(w)32 b(if)i Fs(T)44 b Fw(is)33 b(a)g(\014nite)g Fs(\016)s Fw(-appro)n(ximation)e(to)i Fs(D)r Fw(,)i(then)e Fs(f)9 b Fw(\()p Fs(T)j Fw(\))33 b(is)g(a)440 731 y(\014nite)28 b Fs(")p Fw(-appro)n(ximation)d(to)j Fs(f)9 b Fw(\()p Fs(D)r Fw(\).)p 3244 731 4 57 v 3248 678 50 4 v 3248 731 V 3297 731 4 57 v 440 897 a Fe(Corollary)44 b(8.)j Ft(If)39 b Fs(f)18 b Fw(:)31 b Fs(D)41 b Fl(!)e Fm(R)45 b Ft(is)39 b(a)g(uniformly)h(c)l(ontinuous)e(function)g(on)h(a) g(total)t(ly)440 996 y(b)l(ounde)l(d)30 b(subset)f Fs(D)j Ft(of)f Fm(R)p Ft(,)36 b(then)29 b(the)h(supr)l(emum)f(of)i Fs(f)9 b Fw(\()p Fs(D)r Fw(\))30 b Ft(exists.)565 1162 y Fw(In)25 b(particular,)f(although)h(one)f(cannot)h(exp)r(ect)h(a)e (constructiv)n(e)g(metho)r(d)i(to)f(lo)r(cate)g(a)440 1262 y(p)r(oin)n(t)k(in)f Fs(D)j Fw(at)d(whic)n(h)h Fs(f)37 b Fw(attains)28 b(its)h(suprem)n(um,)f(for)g(ev)n(ery)f Fs(")d(>)h Fw(0)j(there)g(is)g(an)g Fs(x)d Fl(2)g Fs(D)440 1362 y Fw(so)30 b(that)g(sup)14 b Fs(f)9 b Fw(\()p Fs(D)r Fw(\))21 b Fl(\000)f Fs(")27 b(<)g(f)9 b Fw(\()p Fs(x)p Fw(\).)46 b(Compare)29 b(this)i(with)g(Lemma)f(2,)h(and)f(note)g(that)h (the)440 1461 y(existence)25 b(of)g(the)h(maxim)n(um)f(of)g(\014nitely) h(man)n(y)e(reals)g(is)i(a)e(sp)r(ecial)h(case)g(of)g(Corollary)e(8.) 565 1561 y(Note)k(that)h(sequen)n(tial)f(completeness)g(can)g(b)r(e)h (deduced)g(from)f(order)f(completeness)440 1660 y(in)h(a)f(c)n (hoice-free)f(w)n(a)n(y)g([18)o(])1305 1630 y Fu(9)1342 1660 y Fw(.)37 b(Ho)n(w)n(ev)n(er,)25 b(when)h(one)g(literally)g (transfers)f(that)i(deduction)440 1760 y(to)g(the)g(presen)n(t)f(con)n (text,)g(it)i(fails)e(to)h(automatically)e(supply)i(the)g(required)f (mo)r(duli.)37 b(As)440 1860 y(a)29 b(digression,)f(let)h(us)g(th)n(us) h(pro)n(vide)e(a)g(b)r(etter)i(suited)f(argumen)n(t,)g(whic)n(h)g (still)h(do)r(es)e(not)440 1959 y(mak)n(e)f(an)n(y)g(use)g(of)h(coun)n (table)f(c)n(hoice.)440 2125 y Ft(R)l(e)l(duction)i(of)i(se)l(quential) f(to)g(or)l(der)g(c)l(ompleteness.)43 b Fw(Let)32 b(\()p Fs(a)2386 2137 y Fr(m)2450 2125 y Fw(\))2482 2137 y Fr(m)p Fq(2)p Fp(N)2664 2125 y Fw(b)r(e)h(a)f Fv(Ca)n(uchy)j Fw(se-)440 2225 y(quence)26 b(of)g(real)f(n)n(um)n(b)r(ers,)h(and)h (de\014ne)f Fs(L)g Fw(as)f(the)i(set)f(of)g(all)g(the)h Fs(p)22 b Fl(2)i Fm(Q)38 b Fw(for)25 b(whic)n(h)i(there)440 2325 y(is)33 b(an)g Fs(N)41 b Fl(2)33 b Fm(N)43 b Fw(so)32 b(that)i Fs(p)e(<)g(a)1447 2337 y Fr(m)1543 2325 y Fw(for)h(all)g Fs(m)f Fl(\025)g Fs(N)9 b Fw(.)53 b(This)33 b Fs(L)g Fw(clearly)f(is)h(inhabited)h(and)440 2424 y(b)r(ounded)29 b(ab)r(o)n(v)n(e;)e(w)n(e)h(\014rst)h(sho)n(w)e(that)i Fs(L)f Fw(is)g(order)g(lo)r(cated)g(from)g(ab)r(o)n(v)n(e.)38 b(T)-7 b(o)28 b(this)h(end,)440 2524 y(let)f Fs(r)e(<)c(s)28 b Fw(b)r(e)g(rational)e(n)n(um)n(b)r(ers.)37 b(F)-7 b(or)27 b Fs(")22 b Fw(=)1875 2491 y Fr(s)p Fq(\000)p Fr(r)p 1875 2505 116 4 v 1916 2552 a Fu(3)2001 2524 y Fw(,)28 b(w)n(e)f(ha)n(v)n(e)1004 2707 y Fs(a)1048 2722 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))1224 2707 y Fl(\000)18 b Fs(")23 b Fl(\024)f Fs(a)1500 2719 y Fr(m)1586 2707 y Fl(\024)h Fs(a)1718 2722 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))1893 2707 y Fw(+)18 b Fs(")83 b Fw(for)27 b(all)g Fs(m)c Fl(\025)g Fs(M)9 b Fw(\()p Fs(")p Fw(\),)440 2889 y(and)31 b(still)h Fs(r)24 b Fw(+)d Fs(")29 b(<)h(s)21 b Fl(\000)f Fs(")p Fw(,)33 b(so)e(that)h(either)f Fs(r)24 b Fw(+)d Fs(")29 b(<)h(a)2204 2904 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))2392 2889 y Fw(or)31 b Fs(a)2542 2904 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))2728 2889 y Fs(<)32 b(s)21 b Fl(\000)f Fs(")p Fw(.)49 b(In)32 b(the)440 2989 y(former)d(case,)h Fs(r)g(<)d(q)j(<)d(a)1272 3004 y Fr(M)6 b Fu(\()p Fr(")p Fu(\))1449 2989 y Fl(\000)19 b Fs(")30 b Fw(for)g(some)f Fs(q)i Fl(2)c Fm(Q)6 b Fw(,)37 b(for)29 b(whic)n(h)h Fs(q)g Fl(2)e Fs(L)p Fw(.)44 b(In)30 b(the)h(latter)440 3088 y(case,)c Fs(p)c(<)f(s)28 b Fw(for)f(ev)n(ery)f Fs(p)d Fl(2)g Fs(L)p Fw(.)565 3188 y(By)36 b(order)f(completeness,)j Fs(L)e Fw(has)g(a)g(suprem)n(um)g Fs(b)p Fw(.)63 b(T)-7 b(o)36 b(conclude)g(the)h(pro)r(of,)h(w)n(e)440 3288 y(no)n(w)25 b(v)n(erify)h(that)g(\()p Fs(a)1095 3300 y Fr(m)1158 3288 y Fw(\))h(con)n(v)n(erges)c(to)j Fs(b)g Fw(with)h(mo)r(dulus)f Fs(")d Fl(7!)g Fs(M)2528 3220 y Fk(\000)2576 3255 y Fr(")p 2576 3269 34 4 v 2576 3316 a Fu(2)2619 3220 y Fk(\001)2657 3288 y Fw(.)36 b(Giv)n(en)26 b(a)g(rational)440 3387 y Fs(")33 b(>)h Fw(0,)h(pic)n(k)f Fs(p)f Fl(2)h Fs(L)g Fw(with)g Fs(b)23 b Fl(\000)1500 3355 y Fr(")p 1499 3369 V 1499 3416 a Fu(2)1576 3387 y Fs(<)33 b(p)p Fw(,)j(and)d Fs(N)43 b Fl(2)34 b Fm(N)44 b Fw(with)35 b Fs(p)e(<)g(a)2648 3399 y Fr(m)2745 3387 y Fw(for)h(all)g Fs(m)f Fl(\025)g Fs(N)9 b Fw(.)440 3487 y(F)-7 b(or)31 b Fs(K)36 b Fw(=)29 b(max)963 3420 y Fk(\010)1011 3487 y Fs(N)t(;)14 b(M)1209 3420 y Fk(\000)1258 3454 y Fr(")p 1257 3468 V 1257 3516 a Fu(2)1300 3420 y Fk(\001\011)1386 3487 y Fw(,)33 b(observ)n(e)d(\014rst)i(that)g Fs(b)21 b Fl(\000)2255 3454 y Fr(")p 2255 3468 V 2255 3516 a Fu(2)2328 3487 y Fs(<)29 b(a)2466 3499 y Fr(k)2539 3487 y Fw(and)i(th)n(us)h Fs(b)21 b Fl(\000)g Fs(")30 b(<)f(a)3241 3499 y Fr(m)440 3595 y Fw(for)i(all)h Fs(m)f Fl(\025)f Fs(M)980 3528 y Fk(\000)1028 3562 y Fr(")p 1027 3576 V 1027 3623 a Fu(2)1070 3528 y Fk(\001)1122 3595 y Fw(.)50 b(Assume)32 b(next)h(that)f Fs(a)1927 3607 y Fr(k)1998 3595 y Fs(>)e(b)21 b Fw(+)2247 3562 y Fr(")p 2246 3576 V 2246 3623 a Fu(2)2289 3595 y Fw(.)50 b(Then)32 b Fs(a)2627 3607 y Fr(k)2689 3595 y Fl(\000)2786 3562 y Fr(")p 2785 3576 V 2785 3623 a Fu(2)2859 3595 y Fs(>)e(b)p Fw(,)j(so)e(that)440 3694 y Fs(b)k(<)g(p)g(<)g(a)832 3706 y Fr(k)896 3694 y Fl(\000)994 3662 y Fr(")p 994 3676 V 994 3723 a Fu(2)1072 3694 y Fw(for)f(some)g(rational)g Fs(p)p Fw(,)j(for)d(whic)n(h,)j(in)e (particular,)g Fs(p)g(<)g(a)2984 3706 y Fr(m)3082 3694 y Fw(for)g(all)440 3794 y Fs(m)c Fl(\025)f Fs(M)729 3727 y Fk(\000)777 3761 y Fr(")p 776 3775 V 776 3823 a Fu(2)819 3727 y Fk(\001)890 3794 y Fw(and)i(th)n(us)g Fs(p)e Fl(2)h Fs(L)p Fw(,)i(a)f(con)n(tradiction)f(to)h Fs(p)e(>)g(b)p Fw(.)50 b(Hence)33 b Fs(a)2776 3806 y Fr(k)2847 3794 y Fl(\024)d Fs(b)21 b Fw(+)3096 3761 y Fr(")p 3095 3775 V 3095 3823 a Fu(2)3171 3794 y Fw(and)440 3902 y(th)n(us)28 b(also)e Fs(a)833 3914 y Fr(m)919 3902 y Fl(\024)d Fs(b)18 b Fw(+)g Fs(")27 b Fw(for)g(all)h Fs(m)23 b Fl(\025)f Fs(M)1726 3835 y Fk(\000)1775 3869 y Fr(")p 1774 3883 V 1774 3931 a Fu(2)1817 3835 y Fk(\001)1855 3902 y Fw(.)p 3244 3902 4 57 v 3248 3849 50 4 v 3248 3902 V 3297 3902 4 57 v 565 4068 a(Needless)28 b(to)h(sa)n(y)-7 b(,)29 b(the)h(approac)n(h)d(to)i(greatest)f(lo)n(w)n(er)f(b)r(ounds)j(alias)e (in\014ma)h(is)g(com-)440 4168 y(pletely)f(analogous)d(to)j(that)f(of)h (least)f(upp)r(er)h(b)r(ounds)g(alias)e(suprema.)565 4267 y(W)-7 b(e)25 b(next)f(supply)h(the)g(standard)f(constructiv)n(e)f (v)n(ersions)g(of)i(the)g Ft(interme)l(diate)j(value)440 4367 y(the)l(or)l(em)p Fw(.)46 b(A)31 b(function)g Fs(f)18 b Fw(:)29 b([)p Fs(a;)14 b(b)p Fw(])27 b Fl(!)h Fm(R)37 b Fw(is)30 b Ft(lo)l(c)l(al)t(ly)35 b(nonc)l(onstant)i Fw(whenev)n(er)30 b(if)h Fs(a)d Fl(\024)f Fs(a)3188 4337 y Fq(0)3240 4367 y Fs(<)440 4467 y(b)476 4436 y Fq(0)526 4467 y Fl(\024)g Fs(b)j Fw(and)g Fs(d)h Fw(is)f(an)g(arbitrary)e(real,) i(then)g Fs(f)9 b Fw(\()p Fs(c)p Fw(\))28 b Fl(6)p Fw(=)f Fs(d)j Fw(for)g(some)f(real)h Fs(c)d Fl(2)h Fw([)p Fs(a)2922 4436 y Fq(0)2945 4467 y Fs(;)14 b(b)3018 4436 y Fq(0)3041 4467 y Fw(].)44 b(Note)440 4566 y(that)22 b(if)f Fs(f)30 b Fw(is)22 b(con)n(tin)n(uous,)f(then)h(there)f(also)g(is)g(a)g (rational)f Fs(c)h Fw(with)h(that)f(prop)r(ert)n(y)-7 b(.)34 b(Strictly)440 4666 y(monotonous)c(functions)j(are)d(clearly)h (lo)r(cally)g(nonconstan)n(t,)h(and)f(so)h(are)e(nonconstan)n(t)440 4765 y(real)d(p)r(olynomials.)440 4932 y Fe(Prop)s(osition)h(9)h(\(In)m (termediate)g(V)-8 b(alue)29 b(Theorem)f(for)i(Lo)s(cally)g(Nonconstan) m(t)440 5031 y(F)-8 b(unctions\).)45 b Ft(L)l(et)37 b Fs(a)e(<)g(b)i Ft(b)l(e)f(r)l(e)l(al)h(numb)l(ers.)59 b(If)37 b Fs(f)18 b Fw(:)30 b([)p Fs(a;)14 b(b)p Fw(])35 b Fl(!)h Fm(R)43 b Ft(is)37 b(c)l(ontinuous)f(with)440 5131 y Fs(f)9 b Fw(\()p Fs(a)p Fw(\))43 b Fs(<)g Fw(0)g Fs(<)f(f)9 b Fw(\()p Fs(b)p Fw(\))p Ft(,)44 b(and)e(lo)l(c)l(al)t(ly)g (nonc)l(onstant,)h(then)e(we)g(c)l(an)g(\014nd)g Fs(c)i Fl(2)g Fw([)p Fs(a;)14 b(b)p Fw(])41 b Ft(with)440 5230 y Fs(f)9 b Fw(\()p Fs(c)p Fw(\))23 b(=)g(0)p Ft(.)p 440 5304 1146 4 v 532 5358 a Fo(9)567 5381 y Fn(Also)g(the)i(deduction)g (giv)n(en)f(in)g([4])f(can)h(easily)g(b)r(e)g(reform)n(ulated)e (without)j(c)n(hoice.)1851 5662 y Fw(7)p eop %%Page: 8 8 8 7 bop 440 531 a Ft(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)30 b(ma)n(y)f(assume)g(that)h Fs(a;)14 b(b)26 b Fl(2)h Fm(Q)6 b Fw(.)49 b(W)-7 b(e)30 b(construct)f Fs(c;)14 b(d)9 b Fw(:)29 b Fm(N)37 b Fl(!)26 b Fm(Q)42 b Fw(suc)n(h)29 b(that)h(for)440 631 y(all)d Fs(m)970 814 y(a)c Fw(=)f Fs(c)1160 826 y Fu(0)1220 814 y Fl(\024)h Fs(c)1344 826 y Fu(1)1404 814 y Fl(\024)g(\001)14 b(\001)g(\001)23 b(\024)f Fs(c)1735 826 y Fr(m)1822 814 y Fs(<)g(d)1952 826 y Fr(m)2039 814 y Fl(\024)g(\001)14 b(\001)g(\001)23 b(\024)g Fs(d)2377 826 y Fu(1)2437 814 y Fl(\024)g Fs(d)2568 826 y Fu(0)2628 814 y Fw(=)g Fs(b;)423 b Fw(\(4\))970 938 y Fs(f)9 b Fw(\()p Fs(c)1088 950 y Fr(m)1150 938 y Fw(\))24 b Fs(<)e Fw(0)h Fs(<)g(f)9 b Fw(\()p Fs(d)1571 950 y Fr(m)1634 938 y Fw(\))p Fs(;)1509 b Fw(\(5\))970 1111 y Fs(d)1013 1123 y Fr(m)1094 1111 y Fl(\000)18 b Fs(c)1213 1123 y Fr(m)1299 1111 y Fl(\024)1387 1019 y Fk(\020)1447 1055 y Fw(2)p 1447 1092 42 4 v 1447 1168 a(3)1498 1019 y Fk(\021)1548 1036 y Fr(m)1611 1111 y Fw(\()p Fs(b)g Fl(\000)g Fs(a)p Fw(\))p Fs(:)1319 b Fw(\(6\))440 1321 y(Let)31 b Fs(c)628 1333 y Fu(0)665 1321 y Fs(;)14 b(:)g(:)g(:)g(;)g(c)886 1333 y Fr(m)980 1321 y Fw(and)31 b Fs(d)1188 1333 y Fu(0)1225 1321 y Fs(;)14 b(:)g(:)g(:)g(;)g(d)1453 1333 y Fr(m)1547 1321 y Fw(b)r(e)32 b(already)d(constructed)i(suc)n(h)g (that)g(\(6\))g(holds.)47 b(Let)440 1421 y Fs(x)40 b Fw(=)f Fs(c)667 1433 y Fr(m)755 1421 y Fw(+)855 1388 y Fu(1)p 855 1402 34 4 v 855 1450 a(3)898 1421 y Fw(\()p Fs(d)973 1433 y Fr(m)1061 1421 y Fl(\000)25 b Fs(c)1187 1433 y Fr(m)1250 1421 y Fw(\))38 b(and)f Fs(y)42 b Fw(=)d Fs(c)1714 1433 y Fr(m)1803 1421 y Fw(+)1902 1388 y Fu(2)p 1902 1402 V 1902 1450 a(3)1945 1421 y Fw(\()p Fs(d)2020 1433 y Fr(m)2109 1421 y Fl(\000)24 b Fs(c)2234 1433 y Fr(m)2297 1421 y Fw(\).)67 b(By)37 b(assumption)g(w)n(e)h(ha)n(v)n(e) 440 1521 y(a)c(rational)g Fs(z)k Fl(2)d Fw([)p Fs(c)1059 1533 y Fr(m)1122 1521 y Fs(;)14 b(d)1202 1533 y Fr(m)1265 1521 y Fw(])35 b(with)g Fs(f)9 b Fw(\()p Fs(z)t Fw(\))34 b Fl(6)p Fw(=)h(0.)57 b(In)35 b(case)f(0)g Fs(<)g(f)9 b Fw(\()p Fs(z)t Fw(\))35 b(let)g Fs(c)2756 1533 y Fr(m)p Fu(+1)2937 1521 y Fw(=)g Fs(c)3073 1533 y Fr(m)3171 1521 y Fw(and)440 1620 y Fs(d)483 1632 y Fr(m)p Fu(+1)674 1620 y Fw(=)44 b Fs(z)t Fw(,)e(and)e(in)h(case)e Fs(f)9 b Fw(\()p Fs(z)t Fw(\))43 b Fs(<)g Fw(0)d(let)h Fs(c)1922 1632 y Fr(m)p Fu(+1)2112 1620 y Fw(=)j Fs(z)f Fw(and)d Fs(d)2520 1632 y Fr(m)p Fu(+1)2711 1620 y Fw(=)k Fs(d)2863 1632 y Fr(m)2926 1620 y Fw(.)75 b(As)40 b Fs(f)48 b Fw(is)440 1720 y(con)n(tin)n(uous,)25 b Fs(f)9 b Fw(\()p Fs(c)p Fw(\))23 b(=)g(0)g(=)f Fs(f)9 b Fw(\()p Fs(d)p Fw(\))27 b(for)e(the)h(real)f(n)n(um)n(b)r(ers)g Fs(c)e Fw(=)g Fs(d)j Fw(giv)n(en)f(b)n(y)h(the)g(mo)r(dulated)440 1820 y Fv(Ca)n(uchy)k Fw(sequence)d(of)g(rationals)f(\()p Fs(c)1634 1832 y Fr(m)1697 1820 y Fw(\))i(and)g(\()p Fs(d)1994 1832 y Fr(m)2057 1820 y Fw(\).)p 3244 1820 4 57 v 3248 1767 50 4 v 3248 1820 V 3297 1820 4 57 v 565 1986 a(One)22 b(migh)n(t)h(think)h(that)f(dep)r(enden)n(t)h(c)n (hoice)e(has)g(b)r(een)i(used)f(in)g(this)g(pro)r(of:)34 b(whether)440 2085 y Fs(f)9 b Fw(\()p Fs(z)t Fw(\))25 b Fs(>)f Fw(0)29 b(or)f Fs(f)9 b Fw(\()p Fs(z)t Fw(\))24 b Fs(<)h Fw(0)j(ma)n(y)h(w)n(ell)f(dep)r(end)i(on)e(the)h(c)n(hoice)f (of)h Fs(z)t Fw(.)40 b(Ho)n(w)n(ev)n(er,)27 b(this)j(is)e(not)440 2185 y(so,)f(inasm)n(uc)n(h)g(w)n(e)g(can)h(refer)f(to)g(the)h (algorithm)f(con)n(tained)g(in)h(the)g(assumption)f(that)h Fs(f)440 2285 y Fw(is)f(lo)r(cally)g(nonconstan)n(t.)565 2384 y(In)k(Section)h(3,)g(ho)n(w)n(ev)n(er,)f(w)n(e)g(can)g(do)h(the)g (pro)r(of)f(of)g(the)i(fundamen)n(tal)e(theorem)g(of)440 2484 y(algebra)h(only)i(with)h(the)g(follo)n(wing)e(basic)h(to)r(ol)g (at)g(hand.)57 b(Notice)34 b(that)h(clearly)e(ev)n(ery)440 2583 y(real)27 b(p)r(olynomial)g(is)g(uniformly)h(con)n(tin)n(uous)e (on)h([)p Fs(a;)14 b(b)p Fw(].)440 2749 y Fe(Theorem)38 b(10)h(\(Appro)m(ximate)g(In)m(termediate)f(V)-8 b(alue)39 b(Theorem\).)44 b Ft(L)l(et)35 b Fs(a)f(<)g(b)440 2849 y Ft(b)l(e)j(r)l(e)l(al)f(numb)l(ers.)58 b(F)-6 b(or)37 b(every)h(uniformly)f(c)l(ontinuous)f(function)g Fs(f)18 b Fw(:)30 b([)p Fs(a;)14 b(b)p Fw(])35 b Fl(!)g Fm(R)43 b Ft(with)440 2949 y Fs(f)9 b Fw(\()p Fs(a)p Fw(\))23 b Fl(\024)g Fw(0)f Fl(\024)h Fs(f)9 b Fw(\()p Fs(b)p Fw(\))p Ft(,)30 b(and)g(every)h Fs(")23 b(>)f Fw(0)p Ft(,)30 b(we)g(c)l(an)g(\014nd)f Fs(c)23 b Fl(2)h Fw([)p Fs(a;)14 b(b)p Fw(])29 b Ft(such)h(that)f Fl(j)p Fs(f)9 b Fw(\()p Fs(c)p Fw(\))p Fl(j)24 b(\024)e Fs(")p Ft(.)440 3115 y(Pr)l(o)l(of.)43 b Fw(In)32 b(the)h(sequel)e(w)n(e)h(rep)r (eatedly)f(in)n(v)n(ok)n(e)f(Lemma)i(1.)49 b(Let)32 b Fs(")e(>)g Fw(0,)j(and)f(compare)440 3214 y Fs(f)9 b Fw(\()p Fs(a)p Fw(\))27 b(and)g Fs(f)9 b Fw(\()p Fs(b)p Fw(\))26 b(with)i Fl(\000)p Fs(")22 b(<)h Fl(\000)1441 3182 y Fr(")p 1439 3196 34 4 v 1439 3243 a Fu(2)1509 3214 y Fw(and)1680 3182 y Fr(")p 1680 3196 V 1680 3243 a Fu(2)1746 3214 y Fs(<)f(")p Fw(,)27 b(resp)r(ectiv)n(ely)-7 b(.)36 b(If)27 b Fl(\000)p Fs(")c(<)f(f)9 b Fw(\()p Fs(a)p Fw(\))27 b(or)f Fs(f)9 b Fw(\()p Fs(b)p Fw(\))23 b Fs(<)f(")p Fw(,)440 3314 y(then)28 b Fl(j)p Fs(f)9 b Fw(\()p Fs(c)p Fw(\))p Fl(j)23 b Fs(<)g(")k Fw(for)g Fs(c)c Fw(=)g Fs(a)28 b Fw(or)e Fs(c)d Fw(=)g Fs(b)p Fw(;)k(whence)h(w)n(e)f(ma)n(y)g(assume) g(that)1352 3518 y Fs(f)9 b Fw(\()p Fs(a)p Fw(\))23 b Fs(<)g Fl(\000)1697 3462 y Fs(")p 1696 3499 42 4 v 1696 3575 a Fw(2)1830 3518 y(and)2058 3462 y Fs(")p 2057 3499 V 2057 3575 a Fw(2)2131 3518 y Fs(<)g(f)9 b Fw(\()p Fs(b)p Fw(\))p Fs(:)440 3729 y Fw(No)n(w)24 b(pic)n(k)g Fs(\016)j(>)22 b Fw(0)i(so)g(that,)i(for)e(all)g Fs(x;)14 b(y)26 b Fl(2)e Fw([)p Fs(a;)14 b(b)p Fw(],)25 b(if)g Fl(j)p Fs(x)12 b Fl(\000)g Fs(y)s Fl(j)24 b(\024)e Fs(\016)s Fw(,)k(then)f Fl(j)p Fs(f)9 b Fw(\()p Fs(x)p Fw(\))j Fl(\000)g Fs(f)d Fw(\()p Fs(y)s Fw(\))p Fl(j)24 b(\024)e Fs(")p Fw(,)440 3828 y(and)33 b(divide)g([)p Fs(a;)14 b(b)p Fw(])33 b(in)n(to)g Fs(a)f Fw(=)f Fs(a)1445 3840 y Fu(0)1515 3828 y Fs(<)g(a)1655 3840 y Fu(1)1724 3828 y Fs(<)h Fl(\001)14 b(\001)g(\001)32 b Fs(<)g(a)2091 3840 y Fr(m)2186 3828 y Fw(=)g Fs(b)g Fw(suc)n(h)h(that)g Fl(j)p Fs(a)2796 3840 y Fr(i)p Fq(\000)p Fu(1)2931 3828 y Fl(\000)22 b Fs(a)3062 3840 y Fr(i)3089 3828 y Fl(j)32 b(\024)g Fs(\016)s Fw(.)440 3928 y(Compare)21 b(ev)n(ery)g Fs(f)9 b Fw(\()p Fs(a)1130 3940 y Fr(i)1158 3928 y Fw(\))22 b(with)h Fl(\000)1472 3895 y Fr(")p 1471 3909 34 4 v 1471 3956 a Fu(2)1537 3928 y Fs(<)1635 3895 y Fr(")p 1634 3909 V 1634 3956 a Fu(2)1677 3928 y Fw(.)35 b(By)23 b(assumption)e Fs(f)9 b Fw(\()p Fs(a)2420 3940 y Fu(0)2457 3928 y Fw(\))24 b Fs(<)e Fl(\000)2676 3895 y Fr(")p 2675 3909 V 2675 3956 a Fu(2)2740 3928 y Fw(and)2907 3895 y Fr(")p 2906 3909 V 2906 3956 a Fu(2)2972 3928 y Fs(<)h(f)9 b Fw(\()p Fs(a)3186 3940 y Fr(m)3249 3928 y Fw(\);)440 4027 y(whence)28 b(w)n(e)f(can)g(\014nd)h Fs(j)33 b Fw(minimal)28 b(suc)n(h)f(that)1253 4227 y Fs(f)9 b Fw(\()p Fs(a)1379 4239 y Fr(j)1414 4227 y Fw(\))23 b Fs(<)1568 4171 y(")p 1566 4208 42 4 v 1566 4284 a Fw(2)1701 4227 y(and)101 b Fl(\000)2031 4171 y Fs(")p 2029 4208 V 2029 4284 a Fw(2)2104 4227 y Fs(<)22 b(f)9 b Fw(\()p Fs(a)2317 4239 y Fr(j)s Fu(+1)2436 4227 y Fw(\))p Fs(:)440 4437 y Fw(Finally)-7 b(,)30 b(compare)d Fs(f)9 b Fw(\()p Fs(a)1200 4449 y Fr(j)1235 4437 y Fw(\))30 b(with)f Fl(\000)p Fs(")c(<)g Fl(\000)1782 4405 y Fr(")p 1781 4419 34 4 v 1781 4466 a Fu(2)1853 4437 y Fw(and)k Fs(f)9 b Fw(\()p Fs(a)2142 4449 y Fr(j)s Fu(+1)2261 4437 y Fw(\))29 b(with)2524 4405 y Fr(")p 2523 4419 V 2523 4466 a Fu(2)2591 4437 y Fs(<)c(")p Fw(.)42 b(If)29 b Fl(\000)p Fs(")c(<)g(f)9 b Fw(\()p Fs(a)3214 4449 y Fr(j)3249 4437 y Fw(\),)440 4537 y(w)n(e)27 b(ha)n(v)n(e)f Fl(j)p Fs(f)9 b Fw(\()p Fs(a)902 4549 y Fr(j)936 4537 y Fw(\))p Fl(j)24 b Fs(<)e(")p Fw(.)37 b(If)27 b Fs(f)9 b Fw(\()p Fs(a)1409 4549 y Fr(j)s Fu(+1)1528 4537 y Fw(\))23 b Fs(<)g(")p Fw(,)k(w)n(e)g(ha)n(v)n(e)f Fl(j)p Fs(f)9 b Fw(\()p Fs(a)2222 4549 y Fr(j)s Fu(+1)2341 4537 y Fw(\))p Fl(j)23 b Fs(<)g(")p Fw(.)36 b(If)27 b(b)r(oth)h Fs(f)9 b Fw(\()p Fs(a)3009 4549 y Fr(j)3044 4537 y Fw(\))23 b Fs(<)g Fl(\000)3263 4504 y Fr(")p 3261 4518 V 3261 4566 a Fu(2)440 4637 y Fw(and)623 4604 y Fr(")p 622 4618 V 622 4665 a Fu(2)707 4637 y Fs(<)41 b(f)9 b Fw(\()p Fs(a)939 4649 y Fr(j)s Fu(+1)1058 4637 y Fw(\),)42 b(then)d(w)n(e)f(w)n (ould)g(ha)n(v)n(e)g Fl(j)p Fs(f)9 b Fw(\()p Fs(a)2093 4649 y Fr(j)s Fu(+1)2211 4637 y Fw(\))27 b Fl(\000)e Fs(f)9 b Fw(\()p Fs(a)2486 4649 y Fr(j)2521 4637 y Fw(\))p Fl(j)41 b Fs(>)g(")p Fw(,)h(con)n(tradicting)440 4736 y Fl(j)p Fs(a)507 4748 y Fr(j)s Fu(+1)644 4736 y Fl(\000)18 b Fs(a)771 4748 y Fr(j)806 4736 y Fl(j)23 b(\024)g Fs(\016)s Fw(.)p 3244 4736 4 57 v 3248 4683 50 4 v 3248 4736 V 3297 4736 4 57 v 565 4902 a(Unlik)n(e)32 b(the)h(pro)r(ofs)e(of)i(this) g(result)f(w)n(e)g(kno)n(w)g(from)g(the)g(literature,)h(the)g (foregoing)440 5002 y(pro)r(of)d(is)g(en)n(tirely)f(c)n(hoice-free.) 1451 4972 y Fu(10)1565 5002 y Fw(It)h(seems)g(to)g(b)r(e)h(c)n (haracteristic)d(of)i(constructions)f(of)440 5101 y(\(uniquely)i (determined)g(or\))f(appro)n(ximate)f(solutions)h(that)h(they)f(do)h (not)f(ev)n(en)g(require)440 5201 y(coun)n(table)d(c)n(hoice)g([18)o (].)p 440 5254 1146 4 v 502 5308 a Fo(10)567 5331 y Fn(Again,)c(this)h (pro)r(of)f(originates)h(in)g(lecture)g(notes)h(of)e(the)i(second)f (author.)1851 5662 y Fw(8)p eop %%Page: 9 9 9 8 bop 440 531 a Fx(3)135 b(The)44 b(F)-11 b(undamen)l(tal)45 b(Theorem)g(of)h(Algebra)440 713 y Fw(In)38 b(the)g(framew)n(ork)d(of)j (constructiv)n(e)e(analysis)h(sk)n(etc)n(hed)f(ab)r(o)n(v)n(e)h(w)n(e)g (no)n(w)g(giv)n(e)f(a)i(de-)440 813 y(tailed)24 b(pro)r(of)g(of)g(the)h (fundamen)n(tal)g(theorem)e(of)i(algebra,)e(essen)n(tially)g(follo)n (wing)h(Martin)440 912 y Fv(Kneser)37 b Fw([9].)889 882 y Fu(11)1022 912 y Fw(The)g(p)r(oin)n(t)f(is)g(that)h(w)n(e)f(presen)n (t)g(the)g(pro)r(of)g(in)h(a)f(c)n(hoice-free)f(form;)440 1012 y(this)27 b(of)f(course)g(includes)h(all)f(auxiliary)f(lemmata.)36 b(Our)26 b(exp)r(osition)g(is)h(rather)e(detailed,)440 1112 y(b)r(ecause)i({)h(as)f(men)n(tioned)g(in)h(the)g(in)n(tro)r (duction)g({)f(w)n(e)g(w)n(an)n(t)g(to)h(mak)n(e)e(the)j(algorithmic) 440 1211 y(con)n(ten)n(t)23 b(of)g(this)g(pro)r(of)f(as)h(clear)e(as)i (p)r(ossible,)g(k)n(eeping)g(in)g(mind)g(that)h(it)f(later)f(ma)n(y)h (serv)n(e)440 1311 y(as)k(a)g(basis)g(for)g(extracting)g(a)g(program.) 565 1411 y(A)40 b(somewhat)g(di\013eren)n(t)g(presen)n(tation)g(of)46 b Fv(Kneser)p Fw('s)41 b(pro)r(of)f(has)g(b)r(een)h(giv)n(en)e(in)440 1510 y([22)o(].)49 b(Ho)n(w)n(ev)n(er,)31 b(it)h(is)f(stressed)g(in)h ([22)o(,)h(p.437])d(that)i(this)g(pro)r(of)f(\\essen)n(tially)f(rests)h (on)440 1610 y(the)d(axiom)f(of)g(c)n(hoice".)1209 1580 y Fu(12)565 1709 y Fw(As)j(usual,)g(real)g(n)n(um)n(b)r(ers)g(giv)n(e)f (rise)g(to)h(complex)g(n)n(um)n(b)r(ers,)h(whic)n(h)f(in)g(the)h (presen)n(t)440 1809 y(con)n(text)d(are)g(represen)n(ted)f(b)n(y)h (pairs)f(of)i(mo)r(dulated)f Fv(Ca)n(uchy)j Fw(sequences)c(of)i (rationals,)440 1909 y(and)e(th)n(us)h(are)f(also)f(sequen)n(tially)h (complete)h(\(Theorem)f(4\).)565 2008 y(Fix)i(the)i(degree)d Fs(n)f Fl(\025)f Fw(1)j(of)h(the)g(p)r(olynomials)f(to)h(b)r(e)g 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Fu(+1)2036 3799 y Fs(:)1098 b Fw(\(12\))440 3948 y(Indeed,)28 b(for)f Fs(i)c Fl(\025)f Fs(k)1042 3960 y Fr(j)1077 3948 y Fw(,)677 4173 y Fs(y)718 4188 y Fr(k)753 4196 y Fj(j)784 4188 y Fu(\()p Fr(j)s Fu(+1\))978 4173 y Fw(=)1195 4117 y(1)p 1076 4154 279 4 v 1076 4233 a(3)1118 4207 y Fr(k)1153 4215 y Fj(j)1184 4207 y Fu(\()p Fr(j)s Fu(+1\))1365 4173 y Fs(y)1406 4185 y Fr(k)1441 4193 y Fj(j)1499 4173 y Fw(=)1632 4117 y(1)p 1596 4154 112 4 v 1596 4230 a(3)1638 4205 y Fr(k)1673 4213 y Fj(j)1718 4173 y Fs(y)1759 4185 y Fr(k)1794 4193 y Fj(j)1825 4185 y Fr(j)1883 4173 y Fl(\025)1948 4188 y Fu(\()p Fq(\003)p Fu(\))2085 4117 y Fw(1)p 2071 4154 70 4 v 2071 4230 a(3)2113 4206 y Fr(i)2150 4173 y Fs(y)2191 4185 y Fr(ij)2272 4173 y Fw(=)2467 4117 y(1)p 2370 4154 236 4 v 2370 4231 a(3)2412 4207 y Fr(i)p Fu(\()p Fr(j)s Fu(+1\))2616 4173 y Fs(y)2657 4185 y Fr(i)2707 4173 y Fw(=)h Fs(y)2836 4188 y Fr(i)p Fu(\()p Fr(j)s Fu(+1\))3044 4173 y Fw(;)440 4386 y(whence)28 b Fs(k)774 4398 y Fr(j)s Fu(+1)916 4386 y Fl(\024)23 b Fs(k)1047 4398 y Fr(j)1109 4386 y Fw(according)j(to)i(the)g(c)n(hoice)e(of)i Fs(k)2112 4398 y Fr(j)s Fu(+1)2259 4386 y Fw(\(\\least)f(p)r (ossible"\).)565 4486 y Ft(Step)32 b Fw(5.)37 b(Cho)r(ose)27 b Fs(j)i(>)22 b Fw(0)28 b(minimal)g(suc)n(h)f(that)h Fs(k)2096 4498 y Fr(j)s Fq(\000)p Fu(1)2240 4486 y Fw(=)23 b Fs(k)2371 4498 y Fr(j)2429 4486 y Fw(=)g Fs(k)2560 4498 y Fr(j)s Fu(+1)2703 4486 y Fw(=:)f Fs(k)s Fw(.)38 b(Because)26 b(of)440 4586 y(\(12\))h(w)n(e)h(ha)n(v)n(e)1051 4768 y Fs(n)23 b Fl(\025)g Fs(k)1255 4780 y Fq(\000)p Fu(1)1367 4768 y Fl(\025)g Fs(k)1498 4780 y Fu(0)1558 4768 y Fl(\025)g Fs(k)1689 4780 y Fu(1)1750 4768 y Fl(\025)f(\001)14 b(\001)g(\001)23 b(\025)g Fs(k)2088 4780 y Fr(j)s Fq(\000)p Fu(1)2231 4768 y Fw(=)g Fs(k)2362 4780 y Fr(j)2420 4768 y Fw(=)g Fs(k)2551 4780 y Fr(j)s Fu(+1)2670 4768 y Fs(;)440 4951 y Fw(where)g(out)h(of)g(eac)n(h)f(pair)g(of)h(t)n(w)n(o)f (consecutiv)n(e)g Fl(\025)g Fw(\(exept)i(for)e(the)h(\014rst)g(one\))g (at)f(least)h(one)440 5051 y(m)n(ust)k(b)r(e)g Fs(>)p Fw(.)36 b(Hence)28 b(for)f(2)p Fs(i)18 b Fl(\000)g Fw(1)k Fl(\024)h Fs(j)32 b Fw(w)n(e)c(ha)n(v)n(e)e Fs(k)2004 5063 y Fu(2)p Fr(i)p Fq(\000)p Fu(1)2173 5051 y Fl(\024)d Fs(n)18 b Fl(\000)g Fs(i)p Fw(.)565 5150 y Ft(Case)34 b Fw(1.)j Fs(j)c Fw(is)27 b(of)h(the)g(form)f(2)p Fs(i)17 b Fl(\000)h Fw(1.)37 b(Then)1403 5369 y Fs(k)1446 5381 y Fr(j)1504 5369 y Fl(\024)23 b Fs(n)18 b Fl(\000)g Fs(i)23 b Fw(=)f Fs(n)d Fl(\000)2044 5312 y Fs(j)k Fw(+)18 b(1)p 2044 5349 182 4 v 2114 5425 a(2)2258 5369 y Fl(!)1831 5662 y Fw(11)p eop %%Page: 12 12 12 11 bop 1403 531 a Fw(2)p Fs(k)1488 543 y Fr(j)1546 531 y Fl(\024)22 b Fw(2)p Fs(n)c Fl(\000)g Fs(j)23 b Fl(\000)18 b Fw(1)23 b Fl(!)1403 656 y Fs(j)28 b Fl(\024)22 b Fw(2\()p Fs(n)c Fl(\000)h Fs(k)1821 668 y Fr(j)1856 656 y Fw(\))f Fl(\000)g Fw(1)c Fs(:)565 806 y Ft(Case)34 b Fw(2.)j Fs(j)c Fw(is)27 b(of)h(the)g(form)f(2)p Fs(i)p Fw(.)36 b(Then)1336 992 y Fs(k)1379 1004 y Fr(j)1437 992 y Fw(=)23 b Fs(k)1568 1004 y Fr(j)s Fq(\000)p Fu(1)1711 992 y Fl(\024)g Fs(n)18 b Fl(\000)g Fs(i)23 b Fw(=)f Fs(n)d Fl(\000)2252 936 y Fs(j)p 2251 973 42 4 v 2251 1049 a Fw(2)2325 992 y Fl(!)1336 1152 y Fw(2)p Fs(k)1421 1164 y Fr(j)1479 1152 y Fl(\024)j Fw(2)p Fs(n)c Fl(\000)g Fs(j)28 b Fl(!)1336 1277 y Fs(j)g Fl(\024)23 b Fw(2\()p Fs(n)18 b Fl(\000)g Fs(k)1754 1289 y Fr(j)1789 1277 y Fw(\))c Fs(:)440 1427 y Fw(Hence)28 b(generally)e(w)n(e)h(ha)n(v)n(e) 1634 1527 y Fs(j)h Fl(\024)23 b Fw(2\()p Fs(n)18 b Fl(\000)g Fs(k)s Fw(\))p Fs(:)1047 b Fw(\(13\))565 1657 y Ft(Step)34 b Fw(6.)41 b(Let)30 b Fs(r)e Fw(:=)1219 1624 y Fr(t)p 1200 1638 65 4 v 1200 1686 a Fu(3)1233 1669 y Fj(j)1300 1657 y Fl(2)e Fm(Q)5 b Fw(.)48 b(W)-7 b(e)30 b(estimate)f Fl(j)p Fs(a)2056 1669 y Fr(k)2097 1657 y Fl(j)p Fs(r)2159 1627 y Fr(k)2230 1657 y Fw(against)f Fl(j)p Fs(a)2585 1669 y Fu(0)2622 1657 y Fl(j)p Fw(.)42 b(F)-7 b(or)29 b Fs(i)c Fw(=)g(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(n)p Fw(,)440 1757 y(b)n(y)28 b(\(11\))f(and)g Fs(k)f Fw(=)d Fs(k)1092 1769 y Fr(j)1154 1757 y Fw(w)n(e)k(ha)n(v)n(e)f Fl(j)p Fs(a)1534 1769 y Fr(k)1575 1757 y Fl(j)p Fs(r)1637 1726 y Fr(k)1702 1757 y Fw(=)c Fl(j)p Fs(a)1856 1769 y Fr(k)1897 1757 y Fl(j)p Fw(\()1982 1724 y Fr(t)p 1962 1738 V 1962 1785 a Fu(3)1995 1769 y Fj(j)2036 1757 y Fw(\))2068 1726 y Fr(k)2133 1757 y Fl(\025)g(j)p Fs(a)2287 1769 y Fr(i)2315 1757 y Fl(j)p Fw(\()2400 1724 y Fr(t)p 2380 1738 V 2380 1785 a Fu(3)2413 1769 y Fj(j)2454 1757 y Fw(\))2486 1726 y Fr(i)2532 1757 y Fl(\000)17 b Fw(2)p Fs(")22 b Fl(\025)2849 1724 y Fu(1)p 2815 1738 101 4 v 2815 1785 a(3)2848 1769 y Fj(nj)2926 1757 y Fl(j)p Fs(a)2993 1769 y Fr(i)3020 1757 y Fl(j)p Fs(t)3073 1726 y Fr(i)3119 1757 y Fl(\000)17 b Fw(2)p Fs(")p Fw(,)440 1856 y(hence)586 2025 y Fl(j)p Fs(a)653 2037 y Fr(k)693 2025 y Fl(j)p Fs(r)755 1991 y Fr(k)820 2025 y Fl(\025)956 1969 y Fw(1)p 918 2006 118 4 v 918 2082 a(3)960 2058 y Fr(nj)1045 2025 y Fs(m)p Fw(\()p Fs(t)p Fw(\))i Fl(\000)f Fw(2)p Fs(")82 b Fw(for)27 b Fs(m)p Fw(\()p Fs(t)p Fw(\))d(=)f(max)o Fl(f)14 b(j)p Fs(a)2160 2037 y Fr(i)2187 2025 y Fl(j)p Fs(t)2240 1995 y Fr(i)2291 2025 y Fl(j)23 b Fs(i)g Fw(=)f(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(n)g Fl(g)27 b Fw(as)g(b)r(efore)820 2225 y Fl(\025)956 2169 y Fw(1)p 918 2206 V 918 2282 a(3)960 2258 y Fr(nj)1045 2225 y Fl(j)p Fs(a)1112 2237 y Fu(0)1149 2225 y Fl(j)19 b(\000)f Fw(3)p Fs(")122 b Fw(b)n(y)28 b(the)g(c)n(hoice)e(of)i Fs(t)g Fw(in)g(\(9\))p Fs(:)440 2408 y Fw(By)g(\(13\))g(w)n(e)f(ha)n(v)n(e)f Fs(nj)i Fl(\024)23 b Fw(2)p Fs(n)p Fw(\()p Fs(n)18 b Fl(\000)g Fs(k)s Fw(\))23 b Fl(\024)g Fw(2)p Fs(n)1816 2378 y Fu(2)1871 2408 y Fl(\000)18 b Fs(k)30 b Fw(and)e(therefore)1415 2598 y Fl(j)p Fs(a)1482 2610 y Fr(k)1522 2598 y Fl(j)p Fs(r)1584 2564 y Fr(k)1649 2598 y Fl(\025)1846 2542 y Fw(1)p 1747 2579 241 4 v 1747 2656 a(3)1789 2632 y Fu(2)p Fr(n)1863 2616 y Fi(2)1895 2632 y Fq(\000)p Fr(k)1998 2598 y Fl(j)p Fs(a)2065 2610 y Fu(0)2102 2598 y Fl(j)18 b(\000)g Fw(3)p Fs(":)827 b Fw(\(14\))440 2799 y(In)28 b(particular)e Fl(j)p Fs(a)994 2811 y Fr(k)1035 2799 y Fl(j)d Fs(>)g Fw(0)k(b)r(ecause)g(of)h Fs(")22 b(<)1799 2759 y Fq(j)p Fr(a)1855 2767 y Fi(0)1887 2759 y Fq(j)p 1799 2780 108 4 v 1836 2828 a Fu(3)1935 2799 y Fl(\001)2074 2766 y Fu(1)p 1987 2780 208 4 v 1987 2839 a(3)2020 2822 y Fi(2)p Fj(n)2085 2810 y Fi(2)2117 2822 y Fh(\000)p Fi(1)2204 2799 y Fw(.)565 2918 y Ft(Step)34 b Fw(7.)42 b(W)-7 b(e)30 b(\014rst)g(determine)f Fs(z)h Fl(2)c Fm(C)51 b Fw(with)30 b Fl(j)p Fs(z)t Fl(j)c Fw(=)f Fs(r)33 b Fw(suc)n(h)c(that)h Fs(a)2690 2930 y Fu(0)2756 2918 y Fw(and)g Fs(a)2964 2930 y Fr(k)3005 2918 y Fs(z)3048 2888 y Fr(k)3117 2918 y Fw(di\013er)440 3017 y(b)n(y)g(a)h(negativ)n(e) e(factor.)45 b(W)-7 b(rite)31 b Fs(a)1518 3029 y Fu(0)1583 3017 y Fw(=)d Fl(j)p Fs(a)1743 3029 y Fu(0)1780 3017 y Fl(j)p Fs(e)1842 2987 y Fr(ix)1903 2995 y Fi(0)1970 3017 y Fw(and)j Fs(a)2179 3029 y Fr(k)2248 3017 y Fw(=)c Fl(j)p Fs(a)2407 3029 y Fr(k)2448 3017 y Fl(j)p Fs(e)2510 2987 y Fr(ix)2571 2996 y Fj(k)2611 3017 y Fw(,)32 b(and)e(let)h Fs(z)h Fw(=)27 b Fl(j)p Fs(r)r Fl(j)p Fs(e)3240 2987 y Fr(iy)440 3117 y Fw(with)h Fs(y)i Fw(suc)n(h)e(that)g Fs(x)1115 3129 y Fu(0)1171 3117 y Fw(+)18 b Fs(\031)26 b Fw(=)d Fs(x)1462 3129 y Fr(k)1522 3117 y Fw(+)18 b Fs(k)s(y)s Fw(.)565 3217 y(W)-7 b(e)33 b(sho)n(w)f Fl(j)p Fs(a)991 3229 y Fu(0)1050 3217 y Fw(+)21 b Fs(a)1180 3229 y Fr(k)1221 3217 y Fs(z)1264 3186 y Fr(k)1304 3217 y Fl(j)32 b(\024)f(j)p Fs(a)1522 3229 y Fu(0)1559 3217 y Fl(j)22 b(\000)f(j)p Fs(a)1757 3229 y Fr(k)1798 3217 y Fs(z)1841 3186 y Fr(k)1881 3217 y Fl(j)h Fw(+)g(2)p Fs(")p Fw(.)51 b(No)n(w)32 b Fs(a)2406 3229 y Fr(k)2447 3217 y Fs(z)2490 3186 y Fr(k)2562 3217 y Fw(=)f Fl(\000)p Fs(ca)2803 3229 y Fu(0)2873 3217 y Fw(with)i Fs(c)e Fl(2)h Fm(R)p Fw(,)440 3316 y Fs(c)23 b(>)g Fw(0,)k(and)g(b)n(y)h(the)g(c)n (hoice)e(of)i Fs(t)g Fw(in)f(\(9\))h(also)f Fl(j)p Fs(a)1962 3328 y Fr(k)2002 3316 y Fs(z)2045 3286 y Fr(k)2086 3316 y Fl(j)c(\024)f(j)p Fs(a)2286 3328 y Fu(0)2324 3316 y Fl(j)c Fw(+)g Fs(")p Fw(.)37 b(Therefore)983 3488 y Fs(c)p Fl(j)p Fs(a)1086 3500 y Fu(0)1123 3488 y Fl(j)23 b Fw(=)g Fl(j)p Fs(a)1324 3500 y Fr(k)1365 3488 y Fs(z)1408 3454 y Fr(k)1448 3488 y Fl(j)g(\024)g(j)p Fs(a)1649 3500 y Fu(0)1686 3488 y Fl(j)18 b Fw(+)g Fs(")83 b Fw(and)g(0)23 b Fs(<)f(c)h Fl(\024)g Fw(1)18 b(+)2645 3432 y Fs(")p 2601 3469 128 4 v 2601 3545 a Fl(j)p Fs(a)2668 3557 y Fu(0)2705 3545 y Fl(j)2738 3488 y Fs(;)440 3682 y Fw(hence)1048 3833 y Fl(j)p Fs(a)1115 3845 y Fu(0)1170 3833 y Fw(+)g Fs(a)1297 3845 y Fr(k)1338 3833 y Fs(z)1381 3798 y Fr(k)1421 3833 y Fl(j)23 b Fw(=)g Fl(j)p Fs(a)1622 3845 y Fu(0)1659 3833 y Fl(j)c(\001)f(j)p Fw(1)g Fl(\000)g Fs(c)p Fl(j)1467 3957 y Fw(=)23 b Fl(j)p Fs(a)1622 3969 y Fu(0)1659 3957 y Fl(j)14 b Fw(max\(1)k Fl(\000)g Fs(c;)c(c)k Fl(\000)g Fw(1\))1467 4082 y(=)23 b Fl(j)p Fs(a)1622 4094 y Fu(0)1659 4082 y Fl(j)14 b Fw(max\(1)k Fl(\000)g Fs(c;)c Fw(1)k Fl(\000)g Fs(c)g Fw(+)g(2\()p Fs(c)g Fl(\000)g Fw(1\)\))1467 4240 y Fl(\024)23 b(j)p Fs(a)1622 4252 y Fu(0)1659 4240 y Fl(j)14 b Fw(max)1851 4148 y Fk(\020)1900 4240 y Fw(1)k Fl(\000)g Fs(c;)c Fw(1)k Fl(\000)g Fs(c)g Fw(+)g(2)2492 4184 y Fs(")p 2448 4221 V 2448 4297 a Fl(j)p Fs(a)2515 4309 y Fu(0)2552 4297 y Fl(j)2585 4148 y Fk(\021)1467 4421 y Fw(=)23 b Fl(j)p Fs(a)1622 4433 y Fu(0)1659 4421 y Fl(j)p Fw(\(1)c Fl(\000)f Fs(c)p Fw(\))g(+)g(2)p Fs(")1467 4559 y Fw(=)23 b Fl(j)p Fs(a)1622 4571 y Fu(0)1659 4559 y Fl(j)c(\000)f(j)p Fs(a)1851 4571 y Fr(k)1892 4559 y Fs(z)1935 4524 y Fr(k)1975 4559 y Fl(j)g Fw(+)g(2)p Fs(":)440 4709 y Fw(W)-7 b(e)28 b(no)n(w)f(obtain)1161 4855 y Fl(j)p Fs(f)9 b Fw(\()p Fs(z)t Fw(\))p Fl(j)22 b(\024)h(j)p Fs(a)1541 4867 y Fu(0)1597 4855 y Fw(+)18 b Fs(a)1724 4867 y Fr(k)1765 4855 y Fs(z)1808 4821 y Fr(k)1848 4855 y Fl(j)g Fw(+)1994 4777 y Fk(X)1972 4955 y Fr(i)p Fq(6)p Fu(=0)p Fr(;k)2150 4855 y Fl(j)p Fs(a)2217 4867 y Fr(i)2245 4855 y Fs(z)2288 4821 y Fr(i)2314 4855 y Fl(j)1386 5095 y(\024)23 b(j)p Fs(a)1541 5107 y Fu(0)1578 5095 y Fl(j)c(\000)f(j)p Fs(a)1770 5107 y Fr(k)1811 5095 y Fs(z)1854 5061 y Fr(k)1894 5095 y Fl(j)g Fw(+)g(2)p Fs(")g Fw(+)2222 5016 y Fk(X)2200 5195 y Fr(i)p Fq(6)p Fu(=0)p Fr(;k)2378 5095 y Fl(j)p Fs(a)2445 5107 y Fr(i)2472 5095 y Fs(z)2515 5061 y Fr(i)2542 5095 y Fl(j)1386 5335 y Fw(=)23 b Fl(j)p Fs(a)1541 5347 y Fu(0)1578 5335 y Fl(j)c(\000)f(j)p Fs(a)1770 5347 y Fr(k)1811 5335 y Fl(j)p Fs(r)1873 5301 y Fr(k)1933 5335 y Fw(+)g(2)p Fs(")f Fw(+)2220 5256 y Fk(X)2197 5435 y Fr(i)p Fq(6)p Fu(=0)p Fr(;k)2375 5335 y Fl(j)p Fs(a)2442 5347 y Fr(i)2470 5335 y Fl(j)p Fs(r)2532 5301 y Fr(i)2560 5335 y Fs(:)3157 5133 y Fw(\(15\))1831 5662 y(12)p eop %%Page: 13 13 13 12 bop 565 531 a Ft(Step)32 b Fw(8.)37 b(Estimate)27 b(of)1301 469 y Fk(P)1388 556 y Fu(1)p Fq(\024)p Fr(i)g Fw(0)k(there)g(are)f Fs(p)e Fl(2)g Fs(L)i Fw(and)h Fs(q)g Fl(2)d Fs(U)38 b Fw(with)30 b Fs(q)22 b Fl(\000)d Fs(p)25 b(<)h(")p Fw(.)41 b(F)-7 b(or)29 b(the)g(latter)g(obser-)440 2657 y(v)-5 b(ation,)32 b(and)f(for)f(the)i (second)e(part)h(of)g(the)g(follo)n(wing,)g(w)n(e)g(ma)n(y)f(refer)h (to)g([22)o(,)h(Chapter)440 2756 y(5,)27 b(Section)h(5];)f(the)h (\014rst)f(part)h(is)f(routine)g(to)h(v)n(erify)-7 b(.)440 2922 y Fe(Lemma)30 b(13.)440 3089 y Fw(\(a\))42 b Ft(If)30 b Fs(L)f Ft(is)h(an)g(op)l(en)g(subset)f(of)i Fm(Q)41 b Ft(and)31 b Fs(U)38 b Ft(denotes)30 b(the)g(op)l(en)g(interior)h(of)g Fm(Q)f Fl(n)18 b Fs(L)p Ft(,)29 b(then)588 3188 y Fw(\()p Fs(L;)14 b(U)9 b Fw(\))39 b Ft(is)h(cut)f(lo)l(c)l(ate)l(d)i(if)g(and)f (only)h(if)f Fs(L)g Ft(is)g(downwar)l(ds)h(monotone)f(and)g(or)l(der) 588 3288 y(lo)l(c)l(ate)l(d)30 b(fr)l(om)g(ab)l(ove.)435 3454 y Fw(\(b\))43 b Ft(The)30 b(fol)t(lowing)j(items)c(ar)l(e)h(e)l (quivalent)h(for)f(a)g(p)l(air)h Fw(\()p Fs(L;)14 b(U)9 b Fw(\))29 b Ft(of)i(subsets)e(of)i Fm(Q)5 b Ft(:)654 3620 y Fw(\(i\))43 b Fs(L)29 b Ft(is)h(inhabite)l(d,)i Fs(U)38 b Ft(is)30 b(inhabite)l(d,)i(and)e Fw(\()p Fs(L;)14 b(U)9 b Fw(\))30 b Ft(is)g(cut)f(lo)l(c)l(ate)l(d.)631 3753 y Fw(\(ii\))43 b Fs(L)e Ft(is)h(downwar)l(ds)h(monotone,)i Fs(U)51 b Ft(is)42 b(upwar)l(ds)g(monotone,)j(and)e Fw(\()p Fs(L;)14 b(U)9 b Fw(\))41 b Ft(is)784 3852 y(appr)l(oximable.)565 4018 y Fw(The)c(strict)g(partial)g(order)e(of)44 b Fv(Dedekind)37 b Fw(reals)g(is)g(giv)n(en)f(b)n(y)h(\()p Fs(L;)14 b(U)9 b Fw(\))39 b Fs(<)g Fw(\()p Fs(L)3123 3988 y Fq(0)3146 4018 y Fs(;)14 b(U)3249 3988 y Fq(0)3272 4018 y Fw(\))440 4118 y(if)41 b(and)e(only)h(if)g Fs(L)1042 4088 y Fq(0)1092 4118 y Fl(\\)27 b Fs(U)49 b Fw(is)40 b(inhabited.)74 b(By)40 b(w)n(a)n(y)f(of)h(this,)j Fl(\024)p Fw(,)g Fl(6)p Fw(=,)g(and)d(=)f(could)h(b)r(e)440 4218 y(in)n(tro)r(duced)35 b({)f(for)g Fv(Dedekind)h Fw(reals)e(just)j(as)e(for)g(an)n(y)g(other)g (t)n(yp)r(e)h(of)g(reals)f({)g(as)g(the)440 4317 y(negation)f(of)g Fs(<)p Fw(,)i(the)f(disjunction)h(of)e Fs(<)g Fw(and)h Fs(>)p Fw(,)h(and)e(the)h(conjunction)g(of)g Fl(\025)f Fw(and)h Fl(\024)p Fw(,)440 4417 y(resp)r(ectiv)n(ely)-7 b(.)34 b(F)-7 b(or)19 b Fv(Dedekind)h Fw(reals,)g(ho)n(w)n(ev)n(er,)g (all)f(these)h(relations)f(admit)h(of)g(a)g(simpler)440 4516 y(de\014nition:)53 b(namely)-7 b(,)38 b(\()p Fs(L;)14 b(U)9 b Fw(\))36 b Fl(\024)f Fw(\()p Fs(L)1621 4486 y Fq(0)1645 4516 y Fs(;)14 b(U)1748 4486 y Fq(0)1770 4516 y Fw(\))36 b(as)f Fs(L)h Fl(\032)g Fs(L)2199 4486 y Fq(0)2257 4516 y Fw(or,)h(equiv)-5 b(alen)n(tly)e(,)37 b(as)e Fs(U)45 b Fl(\033)36 b Fs(U)3258 4486 y Fq(0)3281 4516 y Fw(;)440 4616 y(\()p Fs(L;)14 b(U)9 b Fw(\))32 b(=)h(\()p Fs(L)883 4586 y Fq(0)906 4616 y Fs(;)14 b(U)1009 4586 y Fq(0)1032 4616 y Fw(\))33 b(as)g Fs(L)f Fw(=)g Fs(L)1448 4586 y Fq(0)1505 4616 y Fw(or,)i(equiv)-5 b(alen)n(tly)e(,)34 b(as)f Fs(U)41 b Fw(=)32 b Fs(U)2490 4586 y Fq(0)2513 4616 y Fw(;)37 b(\()p Fs(L;)14 b(U)9 b Fw(\))32 b Fl(6)p Fw(=)g(\()p Fs(L)3015 4586 y Fq(0)3038 4616 y Fs(;)14 b(U)3141 4586 y Fq(0)3164 4616 y Fw(\))34 b(as)440 4716 y Fs(L)497 4686 y Fq(0)542 4716 y Fl(\\)22 b Fs(U)41 b Fw(or)32 b Fs(L)21 b Fl(\\)i Fs(U)1046 4686 y Fq(0)1101 4716 y Fw(b)r(eing)33 b(inhabited.)53 b(Referring)31 b(to)i([22)o(,)h(Chapter)e(5,)i(Section)f(5])f(for)440 4815 y(further)c(details,)f(so)g(for)g(arithmetic,)h(w)n(e)f(write)g Fm(D)48 b Fw(for)27 b(the)h(set)f(of)34 b Fv(Dedekind)28 b Fw(reals.)565 4915 y(Let)c Fb(R)g Fw(b)r(e)h(an)f(arc)n(himedean)f (ordered)h(and)g(v)-5 b(alued)24 b(Heyting)h(\014eld:)35 b(that)25 b(is,)g(a)f(mo)r(del)440 5015 y(of)30 b(all)f(the)i(axioms)d ({)i(for)f(the)h(momen)n(t,)h(except)f(for)f(order)f(completeness)i({)f (that)h(w)n(ere)p 440 5068 1146 4 v 502 5121 a Fo(14)567 5145 y Fn(Although)g(it)e(ma)n(y)g(th)n(us)h(seem)f(redundan)n(t,)j(w)n (e)e(ha)n(v)n(e)h(c)n(hosen)g(to)f(de\014ne)h(a)e Fd(Dedekind)i Fn(real)e(as)h(a)440 5224 y(pair)e(rather)i(than)f(just)g(as)g(either)h (a)f(lo)n(w)n(er)f(or)h(an)g(upp)r(er)g(cut.)45 b(Among)27 b(our)h(reasons)g(for)g(so)f(doing)i(is)440 5303 y(that)h(pairs)f(are)g (more)e(suited)j(when)f(it)g(comes)g(to)g(considering)h Fa(gener)l(alise)l(d)36 b Fn(real)29 b(n)n(um)n(b)r(ers,)g(i.e.)46 b(not)440 5381 y(necessarily)24 b(lo)r(cated)h Fd(Dedekind)f Fn(cuts,)g(as)g(e.g.)f(in)h([13,)f(20)q(].)1831 5662 y Fw(14)p eop %%Page: 15 15 15 14 bop 440 531 a Fw(listed)41 b(b)n(y)g Fv(Bridges)g Fw(in)g([4].)1394 501 y Fu(15)1540 531 y Fw(\(In)g(compliance)f(with)h (Theorem)f(5)g(ab)r(o)n(v)n(e,)i(w)n(e)f(sa)n(y)440 631 y(that)e(an)n(y)f(suc)n(h)g Fb(R)h Fw(is)f Ft(or)l(der)j(c)l(omplete)47 b Fw(whenev)n(er)38 b(the)h(statemen)n(t)f(of)h(that)g(theorem)440 731 y(is)f(v)-5 b(alid)39 b(for)f(this)h Fb(R)f Fw(in)h(place)f(of)h Fm(R)p Fw(.\))76 b(These)38 b(axioms)f({)i(of)f(course,)i(together)e (with)440 830 y(order)c(completeness)h({)g(em)n(b)r(o)r(dy)h(all)f(the) h(prop)r(erties)f(of)g(real)g(n)n(um)n(b)r(ers,)i(suc)n(h)e(as)g(the) 440 930 y(appro)n(ximate)27 b(splitting)i(principle)f(\(cf.)41 b(Lemma)28 b(1\),)h(that)f(are)g(commonly)g(accepted)g(in)440 1029 y(constructiv)n(e)g(mathematics.)41 b(In)29 b(particular,)f(an)n (y)g(suc)n(h)g Fb(R)h Fw(con)n(tains)f Fm(Q)41 b Fw(as)28 b(an)h(ordered)440 1129 y(sub\014eld)24 b(that)g(is)f Ft(dense)31 b Fw(in)24 b(the)g(sense)f(that)h(for)f(all)g Fs(x;)14 b(y)27 b Fl(2)c Fb(R)g Fw(with)i Fs(x)e(<)g(y)j Fw(there)e(is)f Fs(q)j Fl(2)e Fm(Q)440 1229 y Fw(suc)n(h)j(that)h Fs(x)c(<)e(q)k(<)d(y)s Fw(.)565 1328 y(Both)j(the)h Fv(Dedekind)f Fw(reals)f(and)i(the)f(real)g(n)n(um)n(b)r(ers)g(as)f(de\014ned)i (earlier)e(on)h(in)h(this)440 1428 y(pap)r(er,)40 b(b)n(y)e(w)n(a)n(y)f (of)h(mo)r(dulated)g Fv(Ca)n(uchy)i Fw(sequences)d(of)h(rationals,)i (form)d(a)h(p)r(erfect)440 1528 y(mo)r(del)28 b(of)g(those)g(axioms.)37 b(As)28 b(w)n(e)f(ha)n(v)n(e)g(seen)h(ab)r(o)n(v)n(e)f(and)g(shall)h (remem)n(b)r(er)f(b)r(elo)n(w,)h(this)440 1627 y(includes)g(order)e (completeness.)565 1727 y(If)k Fs(S)h Fl(\032)26 b Fb(R)k Fw(is)f(inhabited,)i(b)r(ounded)f(from)f(ab)r(o)n(v)n(e,)g(and)h(order) e(lo)r(cated)h(from)h(ab)r(o)n(v)n(e,)440 1826 y(then)h Fs(\030)668 1838 y Fr(S)744 1826 y Fw(=)d(\()p Fs(L)926 1838 y Fr(S)973 1826 y Fs(;)14 b(U)1067 1838 y Fr(S)1115 1826 y Fw(\))31 b(is)f(a)g Fv(Dedekind)h Fw(real)e(with)i Fs(L)2174 1838 y Fr(S)2252 1826 y Fw(and)f Fs(U)2473 1838 y Fr(S)2552 1826 y Fw(as)g(the)g(op)r(en)h(in)n(teriors)440 1926 y(of)613 2109 y Fl(f)14 b Fs(p)22 b Fl(2)i Fm(Q)34 b Fl(j)23 b Fs(p)g(<)g(r)30 b Fw(for)d(some)g Fs(r)f Fl(2)e Fs(S)18 b Fl(g)83 b Fw(and)g Fl(f)14 b Fs(q)25 b Fl(2)e Fm(Q)35 b Fl(j)23 b Fs(s)g Fl(\024)g Fs(q)30 b Fw(for)e(all)f Fs(s)c Fl(2)g Fs(S)c Fl(g)14 b Fs(;)440 2291 y Fw(resp)r(ectiv)n(ely)-7 b(.)37 b(In)29 b(particular,)e(w)n(e)g (can)h(assign)f(a)g Fv(Dedekind)h Fw(real)f Fs(\031)s Fw(\()p Fs(x)p Fw(\))j(to)e(eac)n(h)f Fs(x)d Fl(2)h Fb(R)440 2391 y Fw(b)n(y)i(setting)1060 2491 y Fs(\031)s Fw(\()p Fs(x)p Fw(\))d(=)f(\()p Fl(f)14 b Fs(p)22 b Fl(2)i Fm(Q)34 b Fl(j)23 b Fs(p)g(<)g(x)14 b Fl(g)p Fs(;)27 b Fl(f)14 b Fs(q)26 b Fl(2)d Fm(Q)35 b Fl(j)23 b Fs(x)h(<)e(q)17 b Fl(g)d Fw(\))27 b Fs(:)440 2640 y Fw(Then)37 b Fs(\031)s Fw(\()p Fs(q)s Fw(\))i(=)f Fs(q)h Fw(for)d(all)h(rationals)e Fs(q)s Fw(,)k(and)d Fs(\031)s Fw(\()p Fs(x)p Fw(\))k Fs(<)d(\031)s Fw(\()p Fs(x)2354 2610 y Fq(0)2379 2640 y Fw(\))g(precisely)e(when)i Fs(x)i(<)e(x)3257 2610 y Fq(0)3281 2640 y Fw(;)440 2740 y(whence)28 b(w)n(e)f(actually)g(ha)n(v) n(e)f(de\014ned)i(an)f(injectiv)n(e)h(mapping)g Fs(\031)12 b Fw(:)28 b Fb(R)23 b Fl(!)g Fm(D)14 b Fw(.)565 2839 y(In)26 b(other)g(w)n(ords,)f Fb(R)h Fw(is)h(canonically)e(em)n(b)r (edded)h(in)n(to)h Fm(D)14 b Fw(,)32 b(whic)n(h)27 b(rephrases)d(the)j (sa)n(y-)440 2939 y(ing)c(that)h(there)f(are)g Ft(at)j(le)l(ast)31 b Fw(as)23 b(man)n(y)h Fv(Dedekind)g Fw(reals)e(as)h(there)g(are)g (real)f(n)n(um)n(b)r(ers)h(of)440 3039 y(an)28 b(arbitrary)e(kind)i (lik)n(e)g Fb(R)p Fw(.)38 b(The)28 b(question)f(w)n(e)h(address)f(no)n (w)g(is)h(under)g(whic)n(h)g(circum-)440 3138 y(stances)c(there)g(are)f Ft(exactly)32 b Fw(as)24 b(man)n(y)-7 b(,)25 b(or)e(rather)g({)h(in)h (view)f(of)h(the)f(canonical)f(c)n(haracter)440 3238 y(of)28 b Fs(\031)j Fw({)c(when)h Fs(\031)i Fw(is)e(a)f(one-to-one)f (corresp)r(ondence.)565 3337 y(In)f([18)o(],)g(it)h(is)f(pro)n(v)n(ed,) 1272 3307 y Fu(16)1366 3337 y Fw(without)g(in)n(v)n(oking)f(an)n(y)g(c) n(hoice)g(principle,)i(that)f Fb(R)g Fw(is)g(order)440 3437 y(complete)19 b(if)h(and)f(only)f(if)i(a)f(mapping)f Fs(\033)12 b Fw(:)29 b Fm(D)43 b Fl(!)23 b Fb(R)c Fw(exists)f(that)i (preserv)n(es)d Fs(<)i Fw(and)f(op)r(erates)440 3537 y(as)29 b(the)h(iden)n(tit)n(y)f(on)h Fm(Q)5 b Fw(.)49 b(In)30 b(fact,)g(if)g Fs(\030)h Fw(=)25 b(\()p Fs(L;)14 b(U)9 b Fw(\))26 b Fl(2)h Fm(D)14 b Fw(,)36 b(then)30 b Fs(L)c Fl(\032)g Fm(Q)41 b Fw({)29 b(considered)g(as)g(a)440 3636 y(subset)e(of)f Fb(R)g Fw({)h(is)f(inhabited,)h(b)r(ounded)g(from) f(ab)r(o)n(v)n(e,)g(and)g(order)f(lo)r(cated)h(from)h(ab)r(o)n(v)n(e;) 440 3736 y(whence)g(the)g(suprem)n(um)f(of)h Fs(L)g Fw(exists)f(pro)n (vided)g(that)h Fb(R)f Fw(is)h(order)e(complete.)37 b(No)n(w)26 b(it)h(is)440 3836 y(routine)g(to)h(v)n(erify)f(that)1306 4018 y Fs(\033)12 b Fw(:)29 b Fm(D)43 b Fl(!)23 b Fb(R)p Fs(;)41 b(\030)27 b Fw(=)c(\()p Fs(L;)14 b(U)9 b Fw(\))23 b Fl(7!)g Fw(sup)14 b Fs(L)440 4201 y Fw(is)27 b(a)f(mapping)g(with)h Fs(\033)s Fw(\()p Fs(\030)t Fw(\))e Fs(<)d(\033)s Fw(\()p Fs(\030)1507 4171 y Fq(0)1531 4201 y Fw(\))27 b(precisely)f(when)h Fs(\030)g(<)c(\030)2336 4171 y Fq(0)2359 4201 y Fw(,)k(and)g(with)g Fs(\033)s Fw(\()p Fs(q)s Fw(\))d(=)f Fs(q)29 b Fw(for)e(all)440 4301 y Fs(q)34 b Fl(2)d Fm(Q)6 b Fw(.)57 b(Con)n(v)n(ersely)-7 b(,)32 b(if)h(w)n(e)f(are)f(giv)n(en)h(suc)n(h)g(a)g(mapping)g Fs(\033)12 b Fw(:)30 b Fm(D)51 b Fl(!)31 b Fb(R)p Fw(,)i(and)f(if)h Fs(S)j Fl(\032)31 b Fb(R)440 4400 y Fw(is)e(inhabited,)g(b)r(ounded)h (from)e(ab)r(o)n(v)n(e,)g(and)h(order)e(lo)r(cated)i(from)f(ab)r(o)n(v) n(e,)g(then)h Fs(\033)s Fw(\()p Fs(\030)3152 4412 y Fr(S)3202 4400 y Fw(\))g({)440 4500 y(with)f Fs(\030)665 4512 y Fr(S)736 4500 y Fw(as)f(de\014ned)h(b)r(efore)f({)g(is)h(a)f(suprem)n (um)g(of)h Fs(S)5 b Fw(.)565 4599 y(As)39 b(a)h(consequence,)i(the)e Fv(Dedekind)f Fw(reals)g(are)g(order)f(complete:)61 b(the)40 b(iden)n(tit)n(y)440 4699 y(mapping)33 b(on)g Fm(D)53 b Fw(clearly)32 b(meets)i(the)f(sp)r(eci\014cations)g(of)g(suc)n(h)g(a) g Fs(\033)s Fw(.)55 b(Moreo)n(v)n(er,)32 b(if)h Fb(R)g Fw(is)440 4799 y(order)25 b(complete)h(and)g(th)n(us)g Fs(\033)12 b Fw(:)28 b Fm(D)43 b Fl(!)24 b Fb(R)h Fw(can)h(b)r(e)h (de\014ned)f(as)g(ab)r(o)n(v)n(e,)f(it)h(automatically)f(is)440 4898 y(the)c(t)n(w)n(o-sided)f(in)n(v)n(erse)f(of)i Fs(\031)s Fw(,)h(so)e(that)h(if)h(\(and,)g(of)f(course,)f(only)h(if)6 b(\))22 b Fb(R)e Fw(is)h(order)e(complete,)440 4998 y(then)k Fb(R)g Fw(is)f(order)g(isomorphic)f(to)i(the)g Fv(Dedekind)f Fw(reals)g(with)h Fm(Q)34 b Fw(b)r(eing)23 b(in)n(v)-5 b(arian)n(t)21 b(under)p 440 5051 1146 4 v 502 5105 a Fo(15)567 5128 y Fn(There)30 b(and)g(in)g(other)g(places)g(order)g (completeness)g(is)f(called)h(the)h(\\\(constructiv)n(e\))h(least-upp)r (er-)440 5207 y(b)r(ound)27 b(principle".)37 b(See)27 b(also)e([17])h(for)f(a)h(case)g(study)h(of)e(real)h(n)n(um)n(b)r(ers)e (as)i(blac)n(k)h(b)r(o)n(xes)f(that,)h(among)440 5286 y(other)d(things,)g(serv)n(es)g(to)g(justify)f(the)i(c)n(hoice)f(of)g (these)g(axioms.)502 5342 y Fo(16)567 5365 y Fn(This)d(go)r(es)h(bac)n (k)g(to)g(an)g(idea)g(of)f(F)-6 b(red)21 b Fd(Richman)g Fn(\(p)r(ersonal)h(comm)n(unication)f(to)h(the)g(\014rst)f(author\).) 1831 5662 y Fw(15)p eop %%Page: 16 16 16 15 bop 440 531 a Fw(the)20 b(isomorphism.)34 b(So)19 b Fm(D)40 b Fw(can)20 b(legitimately)f(b)r(e)i(c)n(haracterised)c(as)j (the)g(order)f(completion)440 631 y(of)29 b Fm(Q)40 b Fw(inasm)n(uc)n(h)28 b(as)g(the)h(statemen)n(t)g(of)g(Theorem)f(5)g(is) h(the)g(righ)n(t)f(w)n(a)n(y)f(to)i(phrase)f(order)440 731 y(completeness)f({)g(whic)n(h)h(from)f(the)h(constructiv)n(e)f(p)r (ersp)r(ectiv)n(e)g(it)h(de\014nitely)g(is.)565 830 y Fv(Tr)n(oelstra)23 b Fw(and)h Fv(v)-7 b(an)27 b(D)n(alen)c Fw(stated)g(in)h([22)o(,)h(Chapter)e(5,)h(5.12.iii])f(that,)i(roughly) 440 930 y(sp)r(eaking,)h Fm(D)47 b Fw(is)26 b({)h(up)g(to)f(order)g (isomorphism)f({)h(the)h(one)g(and)f(only)g(ordered)g(extension)440 1029 y(of)33 b Fm(Q)44 b Fw(in)34 b(whic)n(h)e(ev)n(ery)g(lo)n(w)n(er)f (cut)j(in)f(the)g(rationals)e(p)r(ossesses)h(a)g(least)h(upp)r(er)g(b)r (ound.)440 1129 y(This)21 b(c)n(haracterisation)d(of)j Fm(D)42 b Fw(turns)21 b(out)g(equiv)-5 b(alen)n(t)21 b(to)f(the)i(one)f(w)n(e)f(ga)n(v)n(e)g(ab)r(o)n(v)n(e,)h(b)n(y)g(w)n (a)n(y)440 1229 y(of)j(order)f(completeness.)35 b(In)24 b(fact,)h(the)g(former)e(follo)n(ws)g(from)h(the)g(latter,)h(b)r (ecause)f(ev)n(ery)440 1328 y(lo)n(w)n(er)k(cut)j(satis\014es)e(the)h (h)n(yp)r(otheses)f(of)h(order)f(completeness:)40 b(according)29 b(to)g(the)i(\014rst)440 1428 y(part)25 b(of)g(Lemma)g(13,)g(it)h(is)f (inhabited,)h(b)r(ounded)g(from)f(ab)r(o)n(v)n(e,)g(and)g(order)f(lo)r 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Ft(Pr)l(o)l(c.)31 b(A)n(mer.)f(Math.)h(So)l(c.)p Fw(,)d(128:2749{2752,)22 b(2000.)1831 5662 y(16)p eop %%Page: 17 17 17 16 bop 482 531 a Fw([7])41 b(Herman)g(Geuv)n(ers,)f(Randy)h(P)n (ollac)n(k,)f(F)-7 b(reek)41 b(Wiedijk,)h(and)f(Jan)g(Zw)n(anen)n (burg.)611 631 y(Theory)52 b(dev)n(elopmen)n(t)h(leading)h(to)f(the)h (fundamen)n(tal)g(theorem)f(of)h(algebra.)611 731 y(Man)n(uscript)27 b(\(automatically)f(generated\),)h(June)h(2000.)482 897 y([8])41 b(Ha)5 b(jime)38 b(Ishihara)e(and)i(P)n(eter)e(Sc)n(h)n (uster.)67 b(A)38 b(constructiv)n(e)f(uniform)h(con)n(tin)n(uit)n(y)611 996 y(theorem.)e Ft(Quart.)29 b(J.)h(Math)p Fw(,)f(53:185{193,)23 b(2002.)482 1162 y([9])41 b(Martin)26 b(Kneser.)35 b(Erg\177)-42 b(anzung)24 b(zu)j(einer)g(Arb)r(eit)g(v)n(on)f(Hellm)n(uth)i(Kneser)g (\177)-44 b(ub)r(er)26 b(den)611 1262 y(F)-7 b(undamen)n(talsatz)27 b(der)g(Algebra.)36 b Ft(Math.)31 b(Z.)p Fw(,)d(177:285{287,)23 b(1981.)440 1428 y([10])41 b(Ra)n(y)19 b(Mines,)h(F)-7 b(red)21 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b(asc)n(hner.)73 b Ft(L)l(ehr)l(gang)42 b(der)g(konstruktiven)e (Mathematik.)k(Dr)l(ei)d(B\177)-42 b(ande)p Fw(.)611 4848 y(Manz)24 b(and)g(H\177)-42 b(older-Pic)n(hler-T)-7 b(empsky)g(,)22 b(2nd)j(ed.)f(of)h(v)n(ols.)f(1,2,)f(ibid.,)i(1995)e (edition,)611 4948 y(1991,)i(1992,)h(1993.)440 5114 y([22])41 b(Anne)35 b(S.)g(T)-7 b(ro)r(elstra)34 b(and)h(Dirk)g(v)-5 b(an)35 b(Dalen.)59 b Ft(Constructivism)37 b(in)f(Mathematics.)611 5214 y(A)n(n)29 b(Intr)l(o)l(duction)p Fw(,)f(v)n(olume)g(121,)f(123)g (of)h Ft(Studies)i(in)g(L)l(o)l(gic)h(and)g(the)g(F)-6 b(oundations)611 5313 y(of)30 b(Mathematics)p Fw(.)39 b(North{Holland,)26 b(Amsterdam,)i(1988.)1831 5662 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF