04fall-final - MAT 371 Advanced Calculus /210 December 15,...

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Unformatted text preview: MAT 371 Advanced Calculus /210 December 15, 2004 Final exam name 2 minor type-ohs corrected 1 2 3 4 5 6 7 30 20 30 40 30 30 30 1. a. State the least upper bound axiom . Include definitions of upper bound and least upper bound . b. State the definitions of open and of closed (subsets of a metric space). Briefly summarize the relationship between these two concepts. c. Briefly summarize the relationship between convergent sequences and Cauchy sequences . d. State the definition of compact . State a theorem that characterizes precisely which subsets of R are compact. State a major theorem that requires compactness and continuity . Bonus. Give counterexamples that show that the theorem is not true without the assumptions of compactness and continuity. 2. In each of the following either prove the statement of give a counterexample. a. If a function f : R 7 R has a limit at a R , then the limit is unique....
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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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