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

$\displaystyle f:[0,\pi]\rightarrow R$

$\displaystyle f_n(x)=\frac{e^xcos(nx)}{\sqrt n}$

Proof: If $\displaystyle x \in [0,\pi]$ then $\displaystyle lim_{n\rightarrow \infty} f_n(x)=0$. So the pointwise limit is 0.

Does it converge uniformly?

Consider $\displaystyle |f_n(x)-f(x)|=\frac{e^x}{\sqrt n}|cos(nx)|\leq \frac{e^\pi}{\sqrt n}< \epsilon$ for n large enough. Thus, it converges uniformly. QED.

Is there anything wrong with this proof?