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Math Help - simple uniform convergence q

  1. #1
    Super Member Anonymous1's Avatar
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    simple uniform convergence q

    Determine directly from the definition whether

     f_{n}(x) = \frac{x}{1 + nx} converges uniformly to 0 on [0,\infty).
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  2. #2
    Member Abu-Khalil's Avatar
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    If x=0,f_n(0)=\left.\frac{x}{1+nx}\right|_{x=0}=0,\fo  rall n\in\mathbb{N}.

    If x\not=0\Rightarrow 0\leq \frac{x}{1+nx}=\frac{1}{\frac{1}{x}+n}\leq \frac{1}{n}.

    Then \sup_{x\in[0,\infty)}\left|\frac{x}{1+nx}-0\right|\leq \frac{1}{n}<\epsilon.
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