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# hw1 - Modern Analysis Homework 1 Due 1[20(3 points Rudin Ch...

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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 . (a) [31] (4 points) Find all discontinuities of f , and note that they form a countable dense set. Show that f is nevertheless Riemann-integrable on every bounded interval 1 .

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hw1 - Modern Analysis Homework 1 Due 1[20(3 points Rudin Ch...

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