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Unformatted text preview: 18,100BC final exam December 18, 2007 NAME: 1. (20 points) If S R , show that the set of isolated points of S is at most countable (use that the set of rational numbers is countable). Solution : Let I be the set of isolated points of S . If s I , let d s = inf x S,x = s  x s  . Then d s > 0. Let B s = { x R  x s  < d s / 2 } . Let r s be a rational number contained in B s . If s, t I, s = t , then  s t  d s and  s t  d t , so  s t  d s 2 + d t 2 , and thus B s B t = 0. So r s = r t . Thus the assignment s r s defines an inclusion of I into the set of rational numbers. This implies that I is at most countable. 1 2 2. Answer the following questions without proofs (each question is 5 points). Consider the following two subsets of the complex plane: A = { z C   z  1 } , and B = { z C   z 1  < 1 } . Equip the complex plane with the usual Euclidean metric, and con sider the sets 1) A 2) B 3) A B 4) A B 5) A B c 6) B A c . Which of the sets 16 are (i) open? Answer: 2,6 (ii) closed? Answer: 1,5 (iii) convex? Answer: 1,2,3 (iv) compact? Answer: 1,5 3 3. (5 points each question) Assume that a positive term series n =1 a n is divergent. Determine, with proofs, whether the following series are always convergent, always divergent, or may be either convergent or divergent, depending on the sequence a n . (i) n =1 a n 1 + a n . (ii) n =1 a n 1 + na n . (iii) n =1 a n 1 + n 2 a n ....
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This note was uploaded on 07/15/2008 for the course MATH 100 taught by Professor Ponce during the Winter '07 term at UCSB.
 Winter '07
 Ponce

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