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# PRINCIPLES OF ANALYSIS I MATH 446

Texas A&M

GPA 3.92

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This 2 page Study Guide was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Study Guide belongs to MATH 446 at Texas A&M University taught by Harold Boas in Fall. Since its upload, it has received 42 views. For similar materials see /class/226004/math-446-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 446 Final Exam Fall 2008 Principles of Analysis I Instructions Please write your solutions on your own paper Explain your reasoning in complete sentences A Section 500 D0 both of these problems A1 In this course you learned various c notions Some of these concepts are a countable b closed c connected d compact and e rst category The Cantor set viewed as a subset of the real numbers with the standard metric has which of these properties Why You may substitute problem 01 for problem A1 if you wish A2 1 i i sinz Cons1der the continuous function f 01 a R de ned by f 7 m ls this continuous function uniformly continuous on the open interval 0 1 Explain why or why not You may substitute problem 02 for problem A2 if you wish B Section 500 and Section 200 D0 two of these problems B1 171 Suppose 0 S x S 1 and fnz 7 when n is a positive integer Discuss 7n 1 convergence of the sequence fn on the closed interval 01 Does this sequence of functions converge pointwise uniformly How do you know B2 In the metric space C0 1 of continuous real valued functions on the closed interval 01 let S be the subset consisting of those continuous functions f such that f0 0 and 7 S x 7y for all z and y ls the set S a compact subset of C0 1 Explain December 5 2008 Page 1 of 2 Dr Boas Math 446 Final Exam Fall 2008 Principles of Analysis I B3 State the following three theorems a the Bolzan07Weierstrass theorem for real numbers b Ho39lder7s inequality for sequences and c the Weierstrass approximation theorem any version B4 Suppose M d and N p are homeomorphic metric spaces If M is com plete must N be complete If M is separable must N be separable Explain C Section 200 D0 both of these problems C1 In this course you learned various 0 notions Some of these concepts are a countable b closed c connected d compact and e rst category The set of sequences of 07s and 17s viewed as a subset of the normed space 00 of bounded sequences has which of these properties Why C2 Let M be the metric space R2 00 the punctured plane equipped with the standard metric inherited from 2 gonsider the continuous func tion f M a R de ned by fy 2ij2 continuous on M Explain why or why not ls this function uniformly D Extra credit optional for both Section 500 and Section 200 For bonus points prove either a the Bernstein equivalence theorem about sets of the same cardinality or b the Baire category theorem for complete metric spaces December 5 2008 Page 2 of 2 Dr Boas

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