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04fall-sampletest1

# 04fall-sampletest1 - S that converges to z 5 a What can you...

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MAT 371 Advanced Calculus Sept 29, 2004 Sample test revised (5c reworded) 1 2 3 4 5 6 20 10 20 15 20 15 1. a. State the supremum (or “least upper bound”) axiom. b. State the axioms for a metric . c. State the definition of converging sequence . d. State the definition of closed set . 2. Using only the field axioms, prove that 3 (defined as 3 = (1 + 1) + 1) is NOT equal to 5 (defined as 5 = (((1 + 1) + 1) + 1) + 1). If impossible explain why. 3. a. Use the definition of convergence of a sequence to show that (i) the constant sequence (0 , 0 , 0 , 0 , . . . ) converges. (ii) the sequence ( a n ) n =1 defined by a n = ( - 1) n does not converge. b. Consider the sequence ( a n ) n =1 of real numbers defined recursively by a 1 = 2 and a n +1 = 1 2 a n + 2 a n . Prove that this sequence converges – you may invoke a previously proven general result, but you must precisely restate it. 1 4. Suppose ( X, d ) is a metric space. Show that if z X is a limit point of a subset S X then there exists a sequence of distinct terms in
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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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