Serge Lang Undergraduate Analysis exercise:
(a) Let $\alpha$ be an irrational number. Let $\epsilon > 0$. Show that there exist integers $m, n$ with $n > 0$ such that $ \left| m\alpha - n \right| < \epsilon$.
(b)In fact, given a positive integer $N$, show that there exist integers $m, n$ and $0 < m \leqq N$ such that $\left| m\alpha - n \right| < 1/N$.
Similar question: An problem of irrational number approximation by rationals