PS3-Solutions - MAT 337 Problem Set 3 Sample Solutions Ian...

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MAT 337 Problem Set 3 Sample Solutions Ian Zwiers - March 8, 2009 This document is for demonstration purposes; it is not a marking key. 4.4 (E) (a) Show the sum of closed set A and compact set B is closed in R n . Suppose A + B is not closed. Then there exists a limit point x R n that is not in A + B . That is, there exists ( a k + b k ) k =1 with a k A and b k B such that a k + b k x / A + B . Since B is compact, and ( b k ) is a sequence in B , there exists a convergent subsequence with lim i →∞ b k i = b B . As ( a k + b k ) is convergent, so too is all its subsequences (and to the same limit). Thus lim i →∞ a k i + b k i = x . By the difference of converging limits, this proves lim i →∞ a k i = x - b (and in particular it exists). As the elements of this sequence belong to A this proves x - b is a limit point of A , which implies x - b A since A is closed. We’ve proven x - b A and b B x A + B , which is a contradiction. Thus our supposition was false, and A + B is closed. (b) Is this true for two compact sets and a closed set? Yes. Let A,B be compact sets, and C closed. Then by part (a), B + C is closed. Again by part (a), A + ( B + C ) is closed. (c) Is this true for two closed sets? No. Consider, A = n s 2 : s Z + o and, B = ± - t 2 - 1 t : t Z + ² . These are closed sets as they contain discrete points with non-zero minimum seperation. (The only convergent sequences in
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PS3-Solutions - MAT 337 Problem Set 3 Sample Solutions Ian...

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