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. Define A={an : n∈N} (the set of values of the sequence {an}). If Ais finite, then at least one of the elements of A, sayx, must be equal toan for infinitely many choices of n. More precisely, Bx ={n∈N: an =x}is infinite. We can then define a convergent subsequence as follows. Pick n1 such that an1 =x. Now, since Bx is infinite, we can choose n2 >n1 such that an2 =x. Continuing in this way, we can define a subsequence {ank}which is constant, equal tox and, thus, converges tox. Suppose now that Ais infinite. Since {an}is a bounded sequence there exist c,d ∈Rsuch that c ≤an ≤d for all n∈N, that is, A⊂[c,d]. We define a sequence of nonempty nested closed bounded intervals as follows. Set I1 = [c,d]. Next consider the two subintervals [c, c+d 2 ] and[ c+d 2 ,d]. Since Ais infinite, at least one of A∩[c, c+d 2 ] or A∩[c+d 2 ,d] is infinite. Let I2 = [c, c+d 2 ] if A∩[c, c+d 2 ] is infinite and I2 = [ c+d 2 ,d] otherwise. Continuing in this way, we construct a nested sequence of nonempty closed bounded intervals {In} such that In ∩Ais infinite 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 ∩Ais infinite. 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. Sayℓ =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. □ Definition 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 Nsuch 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 n→∞ an =a. Then for anyε >0, there exists a positive integer Nsuch 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. □
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