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

1. ## 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?

2. No.