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# hw3 - Homework 3 Math CS 120 Winter 2009 Due on Thursday...

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Homework 3 – Math CS 120, Winter 2009 Due on Thursday, January 29th, 2009 1. Consider a sequence of real numbers { x n } n N R . Prove that if there exists a subsequence of { x n } n N that converges to b R , then lim inf n →∞ x n b lim sup n →∞ x n . 2. Find (a) lim x 0 1 + x - 1 - x x . (b) lim x 1 x 5 - 1 x 4 - 1 . (c) lim x 2 3 x - 3 2 x - 2 . 3. Let f : [ a, b ] R be a continuous function, and consider x 1 , x 2 [ a, b ], x 1 x 2 , y R such that f ( x 1 ) y f ( x 2 ) . Prove that there exists c [ x 1 , x 2 ] such that f ( c ) = y . 4. A point a R n is called a boundary point of the set A R n if every open ball centered at a intersects both A and A c = R n \ A . The boundary of the set A is the set of all its boundary points. Prove that
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