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 N such 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|≤ M for all n ∈ N and, 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 K such that |ank − a| < ε/2 for all k ≥ K. Let N = max{N1,K} and consider nℓ such that ℓ> N. Then for any n ≥ 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 Defnition 2.4.1 that {an} is a Cauchy sequence if and only if for every ε > 0, there exists N ∈ N such that |an+p − an| < ε for all n ≥ N and for all p ∈ N. Defnition 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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