Let $\alpha$ be irrational and $S=\{\{n\alpha\}:n\in \mathbb{Z}\}$.

I proved that for any positive integer $N, \exists m\in \mathbb{Z}$ such that $\{m\alpha\}<\frac{1}{N}.$

But how do I use the above fact to show that for $x\in [0,1]$ and $\forall \varepsilon>0,\;\big((x-\varepsilon, x+\varepsilon)\cap S\big)\neq \varnothing?$

Can anyone help me to answer my question.