204PS12009

204PS12009 - Show that every decreasing sequence of real...

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Econ 204 Summer 2009 Problem Set 1 Due in Lecture Friday, July 31 2009 1. Cardinality For each pair of set A and set B, show that A and B are numerically equivalent. (Hint: Show that there exists a bijection f : A ! B; i.e. f is one to one and onto.) (a) A = ( 1 ; 1) B = ( ; + 1 ) (b) A = [0 ; 1] B = (0 ; 1) (c) A B = A [ C where C 2. Induction Using mathematical induction, show the following: n = 1 ; 2 ; 3 ;::: (a) P n i =1 k i = 1 1 k n k 1 , k 6 = 1 : (b) P 1 i = n ( k 1) k i = k 1 n , k > 1 : (c) P n i =1 1 p i ± p n 3. Bijection Suppose f : X ! Y is a bijection, i.e. f is one to one and onto. Show that for any A;B ² X , f ( A \ B ) = f ( A ) \ f ( B ) : 4. Supremum Property and Completeness Axiom Use the Completeness Axiom to prove that every nonempty set of real numbers which is bounded 5. Limit of Decreasing Sequence
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Unformatted text preview: Show that every decreasing sequence of real numbers that is bounded below converges to its in&mum. (Hint: you can directly use the result of question 4) 6. Metric Space (a) & ( x;y ) = ( 1 if x 6 = y otherwise , prove whether or not it is a metric on R n . (b) & ( x;y ) = P n i =1 j x i & y i j , prove whether or not it is a metric on R n . (c) Suppose ( S 1 ;d 1 ) and ( S 2 ;d 2 ) are metric spaces. Show that ( S 1 ³ S 2 ;& ) is a metric space, where & (( x 1 ; x 2 ) ; ( y 1 ;y 2 )) = max f d 1 ( x 1 ; y 1 ) ;d 2 ( x 2 ;y 2 ) g for all x 1 ;y 1 2 S 1 and all x 2 ; y 2 2 S 2 . 1...
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