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Math Help - Uniform Convergence

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    Uniform Convergence

    Define Fn(x)=exp(-nx) for n=1,2,... . Show that Fn converges uniformly on (1, 00).
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Cairo View Post
    Define Fn(x)=exp(-nx) for n=1,2,... . Show that Fn converges uniformly on (1, 00).
    Claim: f_n(x) converges uniformly to f(x) = 0.

    Proof: Let \epsilon > 0 be given. Choose N \in \mathbb{N} so that N > \ln \frac 1 \epsilon.

    Then for all x \in (1, \infty) and n>N we have |f_n(x) - f(x)| = |e^{-nx}| \le |e^{-n}| < |e^{-N}| < |e^{\ln \epsilon}| = \epsilon

    QED
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