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Math Help - Real Analysis - Uniform Convergence

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    Real Analysis - Uniform Convergence

    c. Prove that if there exists an M > 0 such that |fn| <= M and |gn| <= M for all n Є N, then (fn*gn) does converge uniformly.

    Any help with this would be greatly appreciated. I've been working on it all night and haven't really come up with much of anything.
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  2. #2
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    Quote Originally Posted by ajj86 View Post
    c. Prove that if there exists an M > 0 such that |fn| <= M and |gn| <= M for all n Є N, then (fn*gn) does converge uniformly.

    Any help with this would be greatly appreciated. I've been working on it all night and haven't really come up with much of anything.
    I guess you are told that f_n \to f and g_n\to g converge uniformly on a set S. And you want to show f_ng_n converge uniformly on S too. We will argue that f_ng_n\to fg. Thus, we need to show |f_n(x)g_n(x) - f(x)g(x) | < \epsilon for all x\in S if n\geq N. To prove this notice that |f_n(x)g_n(x) - f(x)g(x)| = |f_n(x)g_n(x) - f_n(x)g(x) + f_n(x)g(x) - f(x)g(x)|.
    And apply the triangle inequality. Here you will use the fact that f_n is bounded.

    (It appears that we also need to know that g is bounded on S. But we do not need to worry since g_n are all bounded it means g is bounded too. )
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