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:
\begin{eqnarray}
\int_0^{\pi} \cos(m x) \cos (nx)\, dx = \delta_{mn} \cdot \frac{\pi}{2}
\end{eqnarray} 
or 
\begin{eqnarray}
\int_0^{\pi} (\cos(m x)- \cos (nx))^2\, dx = \delta_{mn} \cdot \pi
\end{eqnarray} 
