ans2 - ECON 831 Solutions Homework #2 Exercise 1: (i)...

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ECON 831 Solutions Homework #2 Exercise 1: (i) Consider S X : S open in X ⇔ ∀ x S, ² > 0 / N ²,X ( x ) S . (ii) Consider ( x m ) X : x m x,x X ⇔ ∀ ² > 0 , M ² R / d ( x m ,x ) < ² m M ² . Exercise 2: We consider 2 equivalent metrics d 1 and d 2 on X . (i) We start with " ": Consider a sequence { x n } that converges in ( X,d 1 ) towards x X : hence by de nition, d 1 ( x, n ,x ) n →∞ -→ 0 . One needs to show that { x n } converges towards x (the same x!) in ( X,d 2 ) : that is we need to show d 2 ( x n ,x ) n →∞ -→ 0 . By assumption, d 1 and d 2 are equivalent, hence one has: 0 d 2 ( x n ,x ) bd 1 ( x n ,x ) n When n goes to in nity, the RHS goes to 0: hence one can conclude that d 2 ( x n ,x ) 0 . " ": Similar because the roles of d 1 and d 2 can be inverted. (ii) We start with " ": Consider an open subset S X under d 1 . By de nition, for any x S , there exists ² > 0 such that N ², ( X,d
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ans2 - ECON 831 Solutions Homework #2 Exercise 1: (i)...

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