Unformatted text preview: S that converges to z . 5. a. What can you say about unions, intersections, and complements of closed sets? State – without proof – the strongest statements that you know to be true. b. Prove that the intersection of two open sets is open. c. Give an example of a countable collection of closed intervals whose union is the open interval (1 , 1). 6. Suppose that ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are sequences of real numbers that converge to L ∈ R and M ∈ R \{ } , respectively. Show that if b n 6 = 0 for all n then the sequence ± a n b n ² ∞ n =1 converges. 1 For the test also review the pair of sequences n→ a n = (1 + 1 n ) n and n→ b n = (1 + 1 n ) n +1 which are easily shown to be monotonically increasing and decreasing by considering the quotients a n +1 a n and b n +1 b n of successive terms....
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 Spring '07
 thieme
 Calculus, Topology, Real Numbers, Metric space, Topological space, Closed set, Advanced Calculus Sample

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