Prove that $\cos (nx)$ has no sub-sequence converging uniformly Let $f_n(x)=\cos(nx)$ on $\Bbb{R}$. Prove that there is no sub-sequence $f_{n_k}(x)$ converging uniformly in $\Bbb{R}$. I have no clue how to address this problem. I just learned Arzela-Ascoli theorem but it seems not fitting into this problem. Anyone can help? Thank you!

Hint: for any $n \ne m$, there is some $x$ such that $|\cos(nx) - \cos(mx) |\ge 1$.
