s3 - MATH 3283W. Sequences, Series, and Foundations:...

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Unformatted text preview: MATH 3283W. Sequences, Series, and Foundations: Writing Intensive. Spring 2009 Homework 3. Problems and Solutions I. Writing Intensive Part 1 . Let f ( x ) be a continuous function on the segment [0 , 1]. Show that f is uniformly continuous on [0 , 1] , which means: > , > , such that from x,y [0 , 1] and | x- y | < it follows | f ( x )- f ( y ) | < . Solution. Suppose this statement is false. Then > 0 such that > there are x,y [0 , 1] with | x- y | < such that | f ( x )- f ( y ) | . Choose a sequence 0 < j 0 as j , and the corresponding x j ,y j [0 , 1] with | x j- y j | < j such that | f ( x j )- f ( y j ) | . By the BolzanoWeierstrass theorem, the sequence { x j } contains a convergent subsequence: x j k x [0 , 1] as k . Then also y j k x as k . Since f ( x ) is continuous at x , we then have f ( x j k )- f ( y j k ) f ( x )- f ( x ) = 0 , so that | f ( x j k )- f ( y j k ) | < for large enough k. This contradiction shows that the given statement is true....
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s3 - MATH 3283W. Sequences, Series, and Foundations:...

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