204PS12008

# 204PS12008 - Econ 204, Summer/Fall 2008 Problem Set 1 Due...

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Econ 204, Summer/Fall 2008 Problem Set 1 Due in Lecture Friday, August 1 1. Real Numbers Suppose that a, b R . Prove the following simple facts about the real numbers R : (a) If for all ε> 0, | a b |≤ ε ,then a = b . (b) If for all ε> 0, a ε ,then a 0. 2. Vector Spaces Show that the set of all polynomials with coeﬃcients in R , [ n N ∪{ 0 } { a 0 + a 1 z + ··· + a n z n | a i R for all i } , is a vector space over R . 3. Induction Use the principle of mathematical induction to prove the following statements: (a) 1 2 +2 2 + ··· + n 2 = n ( n +1)(2 n +1) 6 . (b) A set with n elements has 2 n subsets (do not forget about the empty set). (c) ± ± n i =1 x i ± ± n i =1 | x i | , for any x i R . 4. Set Theory Suppose that A 1 ,A 2 S . Show that: 1 (a) S \ ( A 1 A 2 )=( S \ A 1 ) ( S \ A 2 ) (b) S \ ( A 1 A 2 )=( S \ A 1 ) ( S \ A 2 ) 5. Metric Spaces For each of the following functions, prove whether or not they generate a metric space when

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## This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at Berkeley.

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204PS12008 - Econ 204, Summer/Fall 2008 Problem Set 1 Due...

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