Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let $\{z_n\}_{n\in\mathbb{N}}\subseteq S^1$ be the sequence defined by $z_n:=z^n=\cos(n\theta)+i\sin(n\theta)$.

Prove that the set of the limit points of $\{z_n\}$ is $S^1$.