SCHWEITZER 2020 Problem 2

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$ .