Prove Uniform Convergence on $\frac{cos(n\pi x))}{n}$ as $n \to \infty$ Trying to prove uniform convergence and having a bit of difficulty working out $$ f_n(x)=\frac{cos(n\pi x)}{n}$$ as  $n \to \infty$ on $x \in[0,1]$. I know that obviously because $n \to \infty$ then the denominator will be $\infty$ and the whole equation will tend $\to 0$. 
However I am trying to prove whether or not this series of $f_n$ is uniformly convergent. So ultimately I am looking to figure out if there is an $x$ value which makes $$f_n(x) \not= 0 $$
I think due to the $\cos$, any $x$ going into it will only come out $\in [-1,1]$, then it would have to $= 0$. Just looking for some confirmation.
Thanks in advance.

Recall that for all $x$, $|\cos(x)|\le 1$.  Therefore, we can write
$$\left|\frac{\cos(n\pi x)}{n}\right|\le \frac1n <\epsilon$$
whenever $n>\frac1\epsilon$.
