hw6 - Homework 6 Math 118B Winter 2010 Due on Thursday...

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Homework 6 – Math 118B, Winter 2010 Due on Thursday, March 11th, 2010 1. Let f n ( x ) = 0 x < 1 n +1 , sin 2 ( π x ) 1 n +1 x 1 n , 0 x > 1 n . Show that { f n } converges to a continuous function, but not uniformly. Use the series f n to show that absolute convergence, even for all x , does not imply uniform convergence. 2. For n = 1 , 2 , 3 , . . . , and x real, de±ne f n ( x ) = x 1 + nx 2 . Show that { f n } converges uniformly to a function f , and that the equation f ( x ) = lim n →∞ f n ( x ) is correct for x n = , but false for x = 0. 3. Let { f n } be a sequence of continuous functions which converges uni- formly to a function f on a set E . Prove that lim n →∞ f n ( x n ) = f ( x ) for every sequence { x n } ⊂ E such that x n x , and x E . 4. Suppose that { f n } , { g n } are de±ned on E , and (a) f n has uniformly bounded partial sums. (b)
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This note was uploaded on 12/27/2011 for the course MATH 118b taught by Professor Garcia during the Fall '09 term at UCSB.

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hw6 - Homework 6 Math 118B Winter 2010 Due on Thursday...

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