Exam2_1.pdf

# If a and b are closed sets then a b is closed a b is

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If A and B are closed sets, then A B is closed, A B is closed. 2.2 Accumulation points Definition 8. A element y is an accumulation point for a set S if any neighborhood of y contains at least one element of S that is not y . i. e for any h the intersection S ( y - h, y + h ) r { y } is not empty. Theorem 9. A set S is closed if and only if any accumulation point of S is in S . Theorem 10. (Bolzano Weierstrass) Any infinite bounded set has at least one accumulation point. Theorem 11. An element y is an accumulation of a set S iff there exists a sequence of distinct elements of S that is convergent to y . Theorem 12. Given a convergent sequence s n of real numbers that takes infinitely many distinct values, the set S = { s n , n N } has exactly one accumulation point, the limit of s n . Theorem 13. Let s n be a bounded sequence, s n has a convergent subsequence. 8

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Introductory Real Analysis Math 327, Autumn 2014 University of Washington c 2014, Dr. F. Dos Reis 2.3 Cauchy sequences Definition 14. s n is a Cauchy sequence if for any positive real , there exist an integer N such that for any p > N and q > N , then | s p - s q | < Theorem 15. A sequence s n is convergent if and only if s n is a Cauchy sequence. 3 Chapter 19 Theorem 16. If a series a n is convergent then lim n →∞ a n = 0. 3.1 Section of nonnegative terms Remark: The series X n =1 1 n is divergent. The series X n =1 1 n 2 is convergent. Theorem 17. Suppose that u n > 0 for every n , then the series u n is convergent iff n X k =1 u k is bounded. Theorem 18. Let a n and b n series with non negative terms such that for all the index greater than some N , 0 6 a n 6 b n . If b n is convergent, then a n is convergent. If a n is divergent, then b n is divergent. Theorem 19. Let a n and b n series with positive terms such that lim n →∞ a n b n = c 6 = 0, then either a n and b n are both conver- gent or a n and b n are both divergent. Theorem 20. (Ratio Test) Let a n and b n series with pos- itive terms such that a n +1 a n 6 b n +1 b n for any n , then If b n is convergent, then a n is convergent.
• Fall '08
• Staff
• Math, Topology, Mathematical analysis, Metric space, Introductory Real Analysis

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