I have to find $\sup_{\mathbb{N}} (\sin n + \cos \sqrt{2}n)$. My attempts were as follows:
I defined a sequence $a_n = (n, \sqrt{2}n)$ and considered it mod $2 \pi$. My goal is to prove that for any small ball $B$ (around the point $\left(\frac{\pi}{2}, 0 \right)$ ) there will be some $a_n$, which belongs to $B$. It is clear that $n \hbox{ (mod 2} \pi)$ is dense, so an infinite number of $a_n$ will belong to the $\left[\frac{\pi}{2} - \varepsilon, \frac{\pi}{2} + \varepsilon \right] \times [0,2\pi]$. Let $a_ {n_k}$ be such sequence. Now I have a problem to prove that from this sequence I can choose other subsequence which will belong to B.
Have you got any hints?