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# hw2 - Modern Analysis Homework 2 Due 1[20(3 points Rudin Ch...

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Modern Analysis, Homework 2 Due July 18, 2011 1. [20] (3 points) Rudin, Ch 7, #18: Let { f m } be a uniformly bounded sequence of functions which are Riemann-integrable on [ a, b ], and for each x [ a, b ], put F n ( x ) = Z x a f n ( t ) dt. Prove that a there exists a subsequence { F n k } which converges uniformly on [ a, b ]. 2. [21] (4 points) Rudin, Ch 7, #19: Let K be a compact metric space and S C ( K ). Prove that S is compact if and only if S is uniformly closed, pointwise bounded, and equicontinuous. ( Hint: If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K .) 3. [27] (4 points) Rudin, Ch 8, #1: Define f ( x ) = e - 1 /x 2 if x 6 = 0 , 0 if x = 0 . Prove that f has derivatives of all orders at x = 0, and that f ( n ) (0) = 0 for all n N . ( Hint: You probably don’t want to actually compute a formula for each f ( n ) ( x ). What form does each take?) 4. [32] Given reals a < b , and 0 < ε < b - a use the previous problem to produce infinitely differentiable functions f such that ...

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hw2 - Modern Analysis Homework 2 Due 1[20(3 points Rudin Ch...

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