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Homework 05

Homework 05 - Q Show there is one and only one way of...

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MATH 74 HOMEWORK 5 Due Friday March 8th 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. (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 { x n } converges with respect to d 2 , then it converges with respect to d 1 . (5) Let f : Q Q be a continuous function with respect to the usual metric on
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Unformatted text preview: Q . Show there is one and only one way of extending it to the reals as a continuous function ˜ f : R → R . (6) Show that a Cauchy sequence is bounded (as a set). Conclude that unbounded sequences cannot converge. (7) Find an example of a bounded set B and a continuous function f : R → R such that f ( B ) is not bounded. (8) Show that R is not compact. (9) Show that a bounded set in R is totally bounded. 1...
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