p7 - a n ,b n ). #3. (14 points). Let X be a metric space...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #7 (due on Wednesday, October 26). 50 points are divided between 4 problems. You can use the following Theorem which was proved in class. Theorem. 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 . #1. (12 points). 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. #2. (12 points, Exercise 30 on p. 46). Show that if R 1 = n =1 F n , where F n are closed subsets of R 1 , then at least one F n contains an open interval (
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Unformatted text preview: a n ,b n ). #3. (14 points). Let X be a metric space of bounded sequences x = { x 1 ,x 2 ,...,x n ,... } with the distance d ( x,y ) := sup n | x n-y n | for x = { x n } , y = { y n } . Show that the set K := { x = { x n } X : | x n | c n = const for all n = 1 , 2 ,... } is compact in X if an only if c n 0 as n . #4. (12 points). If a 1 = 1, and a n +1 = 1 1 + a n for n = 1 , 2 ,..., prove that the sequence { a n } converges and nd its limit. You can use without proof the fact that lim n q n = 0 if | q | < 1 ....
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