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204PS2Solution2009

# 204PS2Solution2009 - Econ 204 Summer 2009 Problem Set 2...

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Econ 204 Summer 2009 Problem Set 2 Solutions 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 ± Solution: (a) Boundary ° ( x; y ) 2 R 2 j x 2 + y 2 = 1 ± . Exterior ° ( x; y ) 2 R 2 j x 2 + y 2 < 1 ± . Closure ° ( x; y ) 2 R 2 j x 2 + y 2 ± 1 ± . (b) Boundary ° ( x; y ) 2 R 2 j x ° y = 3 ± . Exterior ° ( x; y ) 2 R 2 j x ° y 6 = 3 ± : Closure ° ( 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. Solution: We prove that E c is open. Then E is closed by the de°nition. Consider x 2 E c ; where E c = f x 2 R 1 : j x ° a j > 2 g : There are only two cases, x > a + 2 and x < a ° 2 . If x > a + 2 , then there exists " = x ° a ° 2 2 > 0 such that x ° " = x + a +2 2 > a + 2 and x + " > x > a + 2 . So B " ( x ) ³ E c . If x < a ° 2 , then there exists " = a ° 2 ° x 2 > 0 such that x + " = a ° 2+ x 2 < a ° 2 and x ° " < x < a ° 2 : So B " ( x ) ³ E c . Hence 8 x 2 E c , there exists " such that B " ( x ) ³ E c ) E c is open. So E is closed. 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 ´ A 2 ´ A 3 ::: ´ A k ´ ::: Show that if A m is bounded for some m , then \ 1 k =1 A k 6 = ? . Solution: Choose any x k 2 A k , k = 1 ; 2 ; :::; Since A m is bounded, f x k g is bounded. By Bolzano-Weierstrass theorem, there exists x 0 and a subsequence f x k i g of f x k g such that f x k i g ! x 0 . For every k , if k i > k , x k i 2 A k i µ A k . Hence x 0 = lim i !1 x k i 2 A k : Since A k is closed,

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