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Partial Solutions to Homework 05

Partial Solutions to Homework 05 - MATH 74 HOMEWORK 5 Due...

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MATH 74 HOMEWORK 5 Due Friday March 6th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let { x n } n =0 be a Cauchy sequence, and suppose that a subsequence { x n i } i =0 converges. Show that the whole sequence { x n } must converge. (2) Show that x A if and only if inf a A d ( x, a ) = 0. First assume x A . Then x is a limit point of A and we’ve seen there is a sequence { a n } n =0 with a i A and lim n →∞ a n = x . Thus, for any > 0 there is an a n such that d ( x, a n ) < . This shows inf a A d ( x, a ) = 0 since if it were greater than zero, there would be some for which there is no a with d ( a, x ) < . Now assume inf d ( x, a ) = 0. Then for any > 0 there is an a A with d ( x, a ) < . So consider points a n gotten by choosing = 1 /n . This gives a sequence with lim n →∞ a n = x , so that x is a limit point of A . But then x A . (3) Show that if x n x then d ( x n , x m ) 0 as n, m → ∞ . (4) Let d 1 and d 2 be two metrics on the same set X . Further suppose that d 1 ( x, y ) λd 2 ( x, y ) for some positive real number λ . Show that if a sequence
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