EC521 - PS2 - Bo˘azi¸i University gc Department of...

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Unformatted text preview: Bo˘azi¸i University gc 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 d1 (x, y ) = d(x,y ) 1+d(x,y ) for any x, y ∈ X. (a) Show that d1 is a metric on X. (b) Show that if S ⊆ X is open in (X, d), then S is open in (X, d1 ). 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 (xn ) and (yn ) be two sequences in X with lim xn = x and lim yn = y. Show that lim d(xn , yn ) = d(x, y ). 4. Determine whether the following sets are open, closed or neither. (a) {(1/n, 1/n2 ) : n ∈ N} ∪ {(0, 0)} ⊆ R2 (b) {(x, y, x2 y 2 ) : x2 + y 2 < 1} ⊆ R3 ∞ (c) [−n, (n − 1)/n] ⊆ R n=1 ∞ (d) (0, 1/n] ⊆ R n=1 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 (xn ) in S and (xn ) in X \S such that lim xn = x = lim xn . 1 7. Let (X, d) be a metric space where d is the discrete metric. (a) Show that any S ⊆ X is open in (X, d). (b) Show that any function f : X → Y is continuous. 8. Let (X, dX ) and (Y, dY )be two metric spaces, and let f : X → Y be a continuous function. Show that f (Cl(A)) ⊆ Cl(f (A)) for any A ⊆ X. 9. Show that f : (0, ∞) → R with f (x) = 1/x is not uniformly continuous. 10. Let (X, dX ) and (Y, dY )be two metric spaces. Then, a function f : X → Y is called Lipschitz continuous if there exists a K > 0 such that dY (f (x), f (y )) ≤ KdX (x, y ) for any x, y ∈ X . Show that if a function f : X → Y is Lipschitz continuous, then it is also uniformly continuous. 2 ...
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This note was uploaded on 10/25/2011 for the course ECON 501 taught by Professor Zobuz during the Spring '11 term at Istanbul Technical University.

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EC521 - PS2 - Bo˘azi¸i University gc Department of...

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