Limit points of $\cos n$.

Find the limit point of the sequence $\{s_n\}$ given by $s_n=\cos n $.

I know by this post Limit of sequence $s_n = \cos(n)$ that the sequence does not converge. But I don't know how to search those points.

Solution 1:

The set $A=\{n+2\pi k:n,k\in\mathbb{Z}\}$ is dense on $\mathbb{R}$. Given a $y\in[-1,1]$ there existe an $x\in\mathbb{R}$ such that $\cos x=y$. Since $A$ is dense on $\mathbb{R}$ there existe a sequence $s_m=n_m+2\pi k_m$ of elements in $A$ such that $\lim\limits_{m\rightarrow\infty}s_m=x$. Then $$\lim\limits_{m\rightarrow\infty}\cos {n_m}=\lim\limits_{m\rightarrow\infty}\cos(n_m+2\pi k_m)=\cos \left(\lim\limits_{m\rightarrow\infty}(n_m+2\pi k_m)\right)=\cos x=y.$$ This implies that the limits points of $\cos n$ is all point in $[-1,1]$.