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Unformatted text preview: Modern Analysis, Homework 1 Due July 11, 2011 1. [20] (3 points) Rudin, Ch 7, #1: Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. 2. Rudin, Ch 7, #2: If { f n } and { g n } converge uniformly on a set E , (a) [20] (2 points) Prove that { f n + g n } converges uniformly on E . (b) [20] (3 points) If, in addition, { f n } and { g n } are sequences of bounded functions, prove that { f n g n } converges uniformly on E . 3. [25] (2 pts) Rudin, Ch 7, #3: Construct sequences { f n } and { g n } which converge uniformly on some set E but such that { f n g n } does not converge uniformly on E (of course, { f n g n } must converge on E ). 4. Rudin, Ch 7, #10 (mostly): Let { x } denote the fractional part of the real number x (by definition, { x } = x b x c , where b x c is the floor function the greatest integer less than or equal to x ). For real x , define f ( x ) = X n =1 { nx } n 2 ....
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This note was uploaded on 10/18/2011 for the course MATH S4062Q taught by Professor Staff during the Summer '11 term at Columbia.
 Summer '11
 Staff

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