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# EC521 - PS2 - Boazii University gc Department of Economics...

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Bo˘ gazi¸ci University Department of Economics Fall 2011 EC 521 MATHEMATICAL METHODS FOR ECONOMICS Problem Set 2 Due 28.10.2010 1. Let ( X, d ) be a metric space. Define d 1 ( x, y ) = d ( x,y ) 1+ d ( x,y ) for any x, y X. (a) Show that d 1 is a metric on X. (b) Show that if S X is open in ( X, d ) , then S is open in ( X, d 1 ) . 2. Let ( X, d ) be a metric space. Let f : R + R be a concave and strictly increasing function with f (0) = 0 . Show that ( X, f d ) is a metric space. 3. Let ( X, d ) be a metric space. Let ( x n ) and ( y n ) be two sequences in X with lim x n = x and lim y n = y. Show that lim d ( x n , y n ) = d ( x, y ) . 4. Determine whether the following sets are open, closed or neither. (a) { (1 /n, 1 /n 2 ) : n N } ∪ { (0 , 0) } ⊆ R 2 (b) { ( x, y, x 2 y 2 ) : x 2 + y 2 < 1 } ⊆ R 3 (c) S n =1 [ - n, ( n - 1) /n ] R (d) T n =1 (0 , 1 /n ] R 5. Let ( X, d ) be a metric space. Let A, B X. (a) Show that Int ( A ) Int ( B ) = Int ( A B ) (b) Show that Cl ( A ) Cl ( B ) = Cl ( A B ) 6. Let ( X, d ) be a metric space and let S X. Show that x Bd ( S ) if and only if there exist ( x n ) in S and ( x 0 n ) in X \ S such that lim

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EC521 - PS2 - Boazii University gc Department of Economics...

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