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Math Help - how to show this series uni. convergent

  1. #1
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    how to show this series uni. convergent

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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by silversand View Post
    Since |u_n(x)|\leq M_n, then for large enough n,m, we can say that \left|\sum_{i=n}^m u_i(x)\right|\leq\sum_{i=n}^m M_i < \epsilon for all x.

    Letting m\to\infty, we can say that \exists~N such that n>N \implies |u_n(x)-u(x)|<\epsilon, so u_i converges uniformly to u.
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    Weierstrass M-test.
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