JhalyssaWilliams.Analysis.HW3 - Section 3.5 2 a False a set is compact iff it is bounded and closed b True by theorem 3.5.6(Bolzano Weierstrass If a

JhalyssaWilliams.Analysis.HW3 - Section 3.5 2 a False a set...

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Section 3.5 2) a) False, a set is compact iff it is bounded and closed b) True, by theorem 3.5.6 (Bolzano- Weierstrass) If a bounded subset S of contains infinitely many points, then there exists at least one point in that is an accumulation point of S. c) True, theorem 3.5.6 d) False, theorem 3.5.6 e) False, F= and consist of finite set. This implies that it converges to a real number n N and it is compact. 3a) Consider the sequence A n = {3 - 1 n } for all n N . It is contained in the set [1,3) and converges to 3. This implies the subsequence converts to 3 but 3 is not part of the original set. No finite sub covers 3b) The set is not compact because it is not closed. Consider A n = {2 - 1 n , 3 + 1 n }, let U n be the union (2 - 1 n ) U ( 3 + 1 n ). The first set converges to 2 and the second set converges to 3 but 2 and 3 do not belong to the original set. No finite sub covers. 4) Suppose a,b,c,d R. Let A be a compact set from [a,b]. By definition A has an inf that is a, and a sup that is b. Now let B be a compact set from [c,d]. By definition B has an inf that is c,
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  • Fall '08
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  • Topology, Empty set, Metric space, Compact space, greatest lower bound, Bolzano- Weierstrass

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