Introduction to Mathematical Analysis I - Second Edition

47 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 { a n } of real numbers has a convergent subsequence. Proof: Suppose { a n } is a bounded sequence. Define A = { a n : n ∈ N } (the set of values of the sequence { a n } ). If A is finite, then at least one of the elements of A , say x , must be equal to a n for infinitely many choices of n . More precisely, B x = { n ∈ N : a n = x } is infinite. We can then define a convergent subsequence as follows. Pick n 1 such that a n 1 = x . Now, since B x is infinite, we can choose n 2 > n 1 such that a n 2 = x . Continuing in this way, we can define a subsequence { a n k } which is constant, equal to x and, thus, converges to x . Suppose now that A is infinite. First observe there exist c , d ∈ R such that c ≤ a n ≤ d for all n ∈ N , that is, A ⊂ [ c , d ] . We define a sequence of nonempty nested closed bounded intervals as follows. Set I 1 = [ c , d ] . Next consider the two subintervals [ c , c + d 2 ] and [ c + d 2 , d ] . Since A is infinite, at least one of A ∩ [ c , c + d 2 ] or A ∩ [ c + d 2 , d ] is infinite. Let I 2 = [ c , c + d 2 ] if A ∩ [ c , c + d 2 ] is infinite and I 2 = [ c + d 2 , d ] otherwise. Continuing in this way, we construct a nested sequence of nonempty closed bounded intervals { I n } such that I n ∩ A is infinite and the length of I n tends to 0 as n → ∞ . We now construct the desired subsequence of { a n } as follows. Let n 1 = 1. Choose n 2 > n 1 such that a n 2 ∈ I 2 . This is possible since I 2 ∩ A is infinite. Next choose n 3 > n 2 such that a n 3 ∈ I 3 . In this way, we obtain a subsequence { a n k } such that a n k ∈ I k for all k ∈ N . Set I n = [ c n , d n ] . Then lim n → ∞ ( d n − c n ) = 0. We also know from the proof of the Monotone Convergence Theorem (Theorem 2.3.1 ) , that { c n } converges. Say ` = lim n → ∞ c n . Thus, lim n → ∞ d n = lim n → ∞ [( d n − c n )+ c n ] = ` as well. Since c k ≤ a n k ≤ d k for all k ∈ N , it follows from Theorem 2.1.5 that lim k → ∞ a n k = ` . This completes the proof. Definition 2.4.1 (Cauchy sequence). A sequence { a n } of real numbers is called a Cauchy sequence if for any ε > 0, there exists a positive integer N such that for any m , n ≥ N , one has | a m − a n | < ε . Theorem 2.4.2 A convergent sequence is a Cauchy sequence. Proof: Let { a n } be a convergent sequence and let lim n → ∞ a n = a . Then for any ε > 0, there exists a positive integer N such that | a n − a | < ε / 2 for all n ≥ N . For any m , n ≥ N , one has | a m − a n | ≤ | a m − a | + | a n − a | < ε / 2 + ε / 2 = ε . Thus, { a n } is a Cauchy sequence.

RkJQdWJsaXNoZXIy NTc4NTAz