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Unformatted text preview: Solutions for HW 5 Problem 107: 2. Let M =   +   and let > 0. Then there exists N such that sup x  f n ( x ) f ( x )  < 2 M for all n N and sup x  g n ( x ) g ( x )  < 2 M for all n N . Now sup x  f n ( x )+ g n ( x ) f ( x ) g ( x )    sup x  f n ( x ) f ( x )  +   sup x  g n ( x ) g ( x )  < 2 + 2 = for all n N . The product f n g n does not have to converge uniformly to fg . Example: Let X = R and f n ( x ) = g n ( x ) = x + 1 n . Then f n converges uniformly to f , where f ( x ) = x . On the other hand sup x  f n ( x ) 2 x 2  = sup x  2 x n + 1 n 2  2 (put x = n ) for all n . Remark: If both f and g are bounded on X , then also f n and g n will be uniformly bounded from some n on, when they converge uniformly to f , respectively g . Then one can show, by adding and subtracting the term f n g , that also the product f n g n converges uniformly to fg ....
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 Fall '11
 Schep

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