04fall-sampletest1

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

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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 ² . 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 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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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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