We had this famous problem in our calculus book that was an interesting problem for me, and can be a begin point for learning ergodic theory:
$\alpha$ is an irrational number. prove that sequence $(n \alpha)_{n=0}^{\infty}$ is dense on $S^1$ (sequence $(n \alpha)_{n=0}^{\infty}\pmod 1$ is dense in $[0,1]$).