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# s3 - MATH 3283W Sequences Series and Foundations Writing...

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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: ε > 0 , δ > 0 , such that from x, y [0 , 1] and | x - y | < δ it follows | f ( x ) - f ( y ) | < ε. Solution. Suppose this statement is false. Then ε > 0 such that δ > 0 there are x, y [0 , 1] with | x - y | < δ such that | f ( x ) - f ( y ) | ≥ ε. Choose a sequence 0 < δ j 0 as j → ∞ , and the corresponding x j , y j [0 , 1] with | x j - y j | < δ j such that | f ( x j ) - f ( y j ) | ≥ ε. By the Bolzano–Weierstrass theorem, the sequence { x j } contains a convergent subsequence: x j k x 0 [0 , 1] as k → ∞ . Then also y j k x 0 as k → ∞ . Since f ( x ) is continuous at x 0 , we then have f ( x j k ) - f ( y j k ) f ( x 0 ) - f ( x 0 ) = 0 , so that | f ( x j k ) - f ( y j k ) | < ε for large enough k. This contradiction shows that the given statement is true. 2 . From Problem 6 in Homework 1 it follows a n = 1 + 1 n n < e < b n = 1 + 1 n n +1 for all n N .

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