s7 - Math 5615H: Introduction to Analysis I. Fall 2011...

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Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #7. Problems and Solutions. You can use the following Theorems which were discussed in class. Theorem 1. A subset K of a metric space ( X,d ) is compact in X if and only if every infinite subset E K has a limit point in K . Theorem 2. A subset K of a metric space ( X,d ) is compact in X if and only if every sequence { p n } K contains a convergent subsequence p n j p K as j . #1. Let K and F be two disjoint subsets of a metric space ( X,d ). Suppose that K is compact in X , and F is closed in X . Show that there is a constant c > 0, such that d ( x,y ) c > 0 for all x K and y F. Proof. Suppose that the given statement is not true. Then there are sequences { x n } K and { y n } F such that d ( x n ,y n ) < 1 n for all n = 1 , 2 , 3 ,.... Since K is compact, there is a convergent subsequence x n j x * K as j . We also have d ( y n j ,x * ) d ( x n j ,y n j ) + d ( x n j ,x * ) < 1 n j + d ( x n j ,x * ) 0 as j . This implies that x * is a limit point of { y n } F , so that x * F , because F is closed. Then x * K F , in contradiction to our assumption that K and F are disjoint. Therefore, the given statement is true....
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s7 - Math 5615H: Introduction to Analysis I. Fall 2011...

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