I have a task. Prove that $(\cos(n\alpha), \sin(n\alpha))$, with $n\in\mathbb{N}$ is dense on unit circle.($\alpha$ chosen such that our set is infinite ) We want to show, that for every element's neighbourhood there is an element of that that is in the neighborhood. We know that there all points are different (there is no cycle on the circle). But why can we use pigeonhole principle to prove that if we divide the circle in $k$ arcs then there exists two elements which a closer than $\frac{2\pi}{k}$?
EDIT: Thanks everyone. Now everything is clear. This question is [SOLVED]
The basic idea is that $(\{n\alpha\})$ is dense in $(0, 1)$. This is Weyl's equidistribution theorem: https://en.wikipedia.org/wiki/Equidistribution_theorem
Your statement easily follows from this.