204PS22009

204PS22009 - f x ± c g 6 Lipschitz Equivalent Theorem 10.8...

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Econ 204 Summer 2009 Problem Set 2 Due in Lecture Tuesday, August 2 2009 1. Boundary, Exterior and Closure Find the boundary, exterior, and closure of the following sets: (a) ( x;y ) 2 R 2 j x 2 + y 2 > 1 ± (b) ( x;y ) 2 R 2 j x y = 3 ± 2. Closed Set Show that E = x 2 R 1 : j x a 2 ± is a closed set where a is a real number. 3. Intersection of Closed Sets Suppose f A k g is a sequence of non-empty closed sets on R n such that A 1 ² A 2 ² A 3 ::: ² A k ² ::: Show that if A m is bounded for some m , then \ 1 k =1 A k 6 = ? . 4. Uniform Continuity in Euclidean Metric Space ( R n ;d ) is the n-dimentional Euclidean metric space. Suppose E ³ R n d ( x;E ) = inf f d ( x;y ) : y 2 E g (a) Show that E is a closed set if and only if for any x 2 R n , there exists y 2 E , such that d ( x;y ) = d ( x;E ) . f : R n ! R + as f ( x ) = d ( x;E ) . Show that f ( x ) is uniformly continuous. 5. Continuous Function in Euclidean Metric Space ( R n ;d ) is the n-dimentional Euclidean metric space. f : R n ! R 1 is a function. Show that f is countinuous if and only if for every c 2 R 1 , A c and B c are closed sets where A c = f x 2 R n : f ( x ) ´ c g and B c = f x 2 R n :
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Unformatted text preview: f ( x ) ± c g . 6. Lipschitz Equivalent Theorem 10.8 on page 107 of de la Fuente says that all norms on R n are Lipschitz-equivalent to the Euclidean norm. The Theorem is correct, but is the proof correct? (a) Suppose q µ q : ( R n ;d ) ! ( R + ;& ) is a norm on R n . d is the metric generated by the norm, d ( x;y ) = q x & y q . & is the Euclidean metric. Show that q µ q is a continuous function. (Hint: Use the triangle inequality.) (b) Now consider the Euclidean norm q µ q E : R n ! R + . The unit circle on R n is de&ned as C = f x 2 R n : q x q E = 1 g . Show that C is compact. (Hint: Show that C is closed and bounded.) (c) Can we use the result of part a and the extreme-value theorem to prove that that q µ q attains a minimum and a maximum in the set C de&ned in part b? 1...
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.

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