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204PS32009 - open set de&nition of connectedness a...

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Econ 204 Summer 2009 Problem Set 3 Due in Lecture Friday August 7 1. Cauchy Sequence Suppose f x n g 2 R n is a Cauchy sequence. It has a subsequence f x n k g such that lim n k !1 x n k = x . Show that lim n !1 x n = x . 2. Compactness Use the open cover de°nition of compactness to show that the subset n n n 2 +1 ; n = 0 ; 1 ; 2 : : : o of R is compact. 3. Completeness a. Show that (0 ; 1) is not complete in the Euclidean metric. b. Show that f n p 2 m ; m 6 = 0 ; n; m 2 N g is not complete in the Euclidean metric. 4. Completeness and Compactness ( R ; d ) is a metric space where d is de°ned as follows: d ( x; y ) = ° 1 if x 6 = y 0 if x = y (a) Show that ( R ; d ) is complete. (b) Is ( R ; d ) bounded? Is ( R ; d ) compact? Prove your answer. 5. Continuous Function Let f : X ! Y be a continuous function ( X and Y are metric spaces). Using the characterization of continuous functions in terms of open sets, the open cover de°nition of compactness, and the
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Unformatted text preview: open set de&nition of connectedness: a. Prove that if X is compact then f ( X ) , the image of f , is compact. b. Prove that if X is connected then f ( X ) , the image of f , is connected. 6. Upper Hemicontinuous Let F : C & R p ! R 1 be a continuous function, where C ± R 1 . Let &( ! ) = f x 2 R n : F ( x;! ) = g be a correspondence. Show directly from the de&nition that if C is compact, then & is an upper hemicontinuous correspondence. (Hint: The proof is by contradiction. Suppose that & is not upper hemicontinuous at some ! ; this tells you that there is a sequence ! n ! ! with certain properties.) 1...
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