Exercise 18, § 1

Let $\alpha$ be an irrational number, F a finite set of noninteger real numbers. For every integer $m > 0$, put $(m\alpha) = m\alpha - [m\alpha]$. Prove that there exist infinitely many integers $m > 0$ such that $[(m\alpha) + r] = [r]$ for every $r \in F$, and infinitely many integers $m > 0$ such that $[(m\alpha) + r] = [r] + 1$ for every $r \in F$.