52 2.4 The Bolzano-Weierstrass Theorem Theorem 2.4.3 A Cauchy sequence is bounded. Proof: Let {an}be a Cauchy sequence. Then for ε =1, there exists a positive integer Nsuch that |am−an| <1 for all m,n≥N. In particular, |an −aN| <1 for all n≥N. Let M=max{|a1|, . . . ,|aN−1|,1+|aN|}. Then, for n =1, . . . ,N−1, we clearly have |an| ≤M. Moreover, for n≥N, |an| =|an −aN+aN| ≤ |an −aN| +|aN| ≤1+|aN| ≤M. Therefore, |an| ≤Mfor all n∈Nand, thus, {an}is bounded. □ Lemma 2.4.4 A Cauchy sequence that has a convergent subsequence is convergent. Proof: Let {an} be a Cauchy sequence that has a convergent subsequence. For any ε >0, there exists a positive integer N1 such that |am−an| <ε/2 for all m,n≥N1. Let {ank}be a subsequence of {an}that converges to some point a. For the above ε, there exists a positive number Ksuch that |ank −a| <ε/2 for all k ≥K. Let N=max{N1,K}and consider nℓ such that ℓ >N. Then for anyn≥N, we have |an −a| ≤ |an −anℓ| +|anℓ −a| <ε/2+ε/2=ε. Therefore, {an}converges to a. □ Theorem 2.4.5 Any Cauchy sequence of real numbers is convergent. Proof: Let {an} be a Cauchy sequence. Then it is bounded by Theorem 2.4.3. By the BolzanoWeierstrass theorem, {an} has a convergent subsequence. Therefore, it is convergent by Lemma 2.4.4. □ Remark 2.4.1 It follows from Definition 2.4.1 that {an} is a Cauchy sequence if and only if for everyε >0, there exists N∈Nsuch that |an+p −an| <ε for all n≥Nand for all p∈N. Definition 2.4.2 A sequence {an}is called contractive if there exists k ∈[0,1) such that |an+2 −an+1| ≤k|an+1 −an| for all n∈N. Theorem 2.4.6 Every contractive sequence is convergent.
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