Introduction to Mathematical Analysis I - Second Edition

48 2.4 THE BOLZANO-WEIERSTRASS THEOREM Theorem 2.4.3 A Cauchy sequence is bounded. Proof: Let { a n } be a Cauchy sequence. Then for ε = 1, there exists a positive integer N such that | a m − a n | < 1 for all m , n ≥ N . In particular, | a n − a N | < 1 for all n ≥ N . Let M = max {| a 1 | , . . . , | a N − 1 | , 1 + | a N |} . Then, for n = 1 , . . . , N − 1, we clearly have | a n | ≤ M . Moreover, for n ≥ N , | a n | = | a n − a N + a N | ≤ | a n − a N | + | a N | ≤ 1 + | a N | ≤ M . Therefore, | a n | ≤ M for all n ∈ N and, thus, { a n } is bounded. Lemma 2.4.4 A Cauchy sequence that has a convergent subsequence is convergent. Proof: Let { a n } be a Cauchy sequence that has a convergent subsequence. For any ε > 0, there exists a positive integer N such that | a m − a n | ≤ ε / 2 for all m , n ≥ N . Let { a n k } be a subsequence of { a n } that converges to some point a . For the above ε , there exists a positive number K such that | a n k − a | < ε / 2 for all k ≥ K . Thus, we can find a positive integer n ` > N such that | a n ` − a | < ε / 2 . Then for any n ≥ N , we have | a n − a | ≤ | a n − a n ` | + | a n ` − a | < ε . Therefore, { a n } converges to a . Theorem 2.4.5 Any Cauchy sequence of real numbers is convergent. Proof: Let { a n } be a Cauchy sequence. Then it is bounded by Theorem 2.4.3 . By the Bolzano- Weierstrass theorem, { a n } has a convergent subsequence. Therefore, it is convergent by Lemma 2.4.4 . Remark 2.4.6 It follows from Definition 2.4.1 that { a n } is a Cauchy sequence if and only if for every ε > 0, there exists N ∈ N such that | a n + p − a n | < ε for all n ≥ N and for all p ∈ N . Definition 2.4.2 A sequence { a n } is called contractive if there exists k ∈ [ 0 , 1 ) such that | a n + 2 − a n + 1 | ≤ k | a n + 1 − a n | for all n ∈ N .