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Math Help - Can someone verify this proof of unif. conv.?

  1. #1
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    Can someone verify this proof of unif. conv.?

    I'm trying to prove whether or not this function converges pointwise and/or uniformly.

    f:[0,\pi]\rightarrow R
    f_n(x)=\frac{e^xcos(nx)}{\sqrt n}

    Proof: If x \in [0,\pi] then lim_{n\rightarrow \infty} f_n(x)=0. So the pointwise limit is 0.
    Does it converge uniformly?
    Consider |f_n(x)-f(x)|=\frac{e^x}{\sqrt n}|cos(nx)|\leq \frac{e^\pi}{\sqrt n}< \epsilon for n large enough. Thus, it converges uniformly. QED.

    Is there anything wrong with this proof?
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