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Unformatted text preview: Fall 2007 ARE211 Problem Set #03 Third Analysis Problem Set Due date: Sep 25 Problem 1 For the following problem, consider an arbitrary universe X and an arbitrary metric d defined on X × X . State whether the following statements are true or false. If they are true, give a proof. If they are wrong, give a counterexample a) If a sequence x n converges then every subsequence of x n must also converge. b) If every subsequence of the sequence x n converges, then x n must also converge. c) If every subsequence of the sequence x n converges, then they all converge to the same point. d) If one subsequence of the sequence x n converges, then x n must also converge. Problem 2 For the following problem, consider an arbitrary universe X and an arbitrary metric d defined on X × X . You are given a sequence x n . Consider the set S = { x n } ∞ n =1 (The set consists of all elements in the sequence). Prove that if b is an accumulation point of the set S, then some subsequence of x n converges to b.converges to b....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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