51 2.4 The Bolzano-Weierstrass Theorem The Bolzano-Weierstrass Theorem is at the foundation of many results in analysis. It is, in fact, equivalent to the completeness axiom of the real numbers. Theorem 2.4.1 — Bolzano-Weierstrass. Every bounded sequence {an} of real numbers has a convergent subsequence. Proof: Suppose {an} is a bounded sequence. Defne A = {an : n ∈ N} (the set of values of the sequence {an}). If A is fnite, then at least one of the elements of A, say x, must be equal to an for infnitely many choices of n. More precisely, Bx = {n ∈ N : an = x} is infnite. We can then defne a convergent subsequence as follows. Pick n1 such that an1 = x. Now, since Bx is infnite, we can choose n2 > n1 such that an2 = x. Continuing in this way, we can defne a subsequence {ank } which is constant, equal to x and, thus, converges to x. Suppose now that A is infnite. Since {an} is a bounded sequence there exist c,d ∈ R such that c ≤ an ≤ d for all n ∈ N, that is, A ⊂ [c,d]. We defne a sequence of nonempty nested closed bounded intervals as follows. Set I1 =[c,d]. c+d d c+d Next consider the two subintervals [c, 2 ] and [ c+ 2 ,d]. Since A is infnite, at least one of A∩[c, 2 ] = [c+d or A ∩ [c+d ,d] is infnite. Let I 2 = [c, c+d ] if A ∩ [c, c+d ] is infnite and I 2 ,d] otherwise. 2 2 2 2 Continuing in this way, we construct a nested sequence of nonempty closed bounded intervals {In} such that In ∩ A is infnite and the length of In tends to 0 as n → ∞. We now construct the desired subsequence of {an} as follows. Let n1 = 1. Choose n2 > n1 such that an2 ∈ I2. This is possible since I2 ∩ A is infnite. Next choose n3 > n2 such that an3 ∈ I3. In this way, we obtain a subsequence {ank } such that ank ∈ Ik for all k ∈ N. Set In =[cn,dn]. Then limn →∞(dn − cn)= 0. We also know from the proof of the Monotone Convergence Theorem (Theorem 2.3.1), that {cn} converges. Say ℓ = limn →∞ cn. Thus, limn →∞ dn = limn →∞[(dn − cn)+ cn]= ℓ as well. Since ck ≤ ank ≤ dk for all k ∈ N, it follows from Theorem 2.1.2 that limk →∞ ank = ℓ. This completes the proof. □ Defnition 2.4.1 (Cauchy sequence). A sequence {an} of real numbers is called a Cauchy sequence if for any ε > 0 there exists a positive integer N such that |am − an| < ε for any m,n ≥ N. Theorem 2.4.2 A convergent sequence is a Cauchy sequence. Proof: Let {an} be a convergent sequence and let lim an = a. n→∞ Then for any ε > 0, there exists a positive integer N such that |an − a| < ε/2 for all n ≥ N. For any m,n ≥ N, one has |am − an|≤|am − a| + |an − a| < ε/2 + ε/2 = ε. Thus, {an} is a Cauchy sequence. □
RkJQdWJsaXNoZXIy NTc4NTAz