I want to prove this below:
(1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$.
I got a hint from somewhere to prove this below:
(2) For any irrational number $\alpha$ and any positive integer $n$, there exist positive integers $k,m$ such that $\left| {\alpha - \frac{m}{k}} \right| < \frac{1}{{{kn}}}$, where $k \leq n$.
I tried to prove (2), but still can't find out how to deal with (1).
Can you help me?