Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that
$$|x- \frac{m}{n} |<\frac{1}{n^2}$$
It's clear that $-1<1/n^2 <1$ with $n \in \mathbb{Z}$. What I aim to prove is that given a irrational number $x$, I can find infinitely many rational numbers of the form $m/n$, such that the distance between $x$ and $m/n$ is less than the distance from $1/n^2$ and $0$.
With the Archimedian property of the Real numbers I found the following inequalities for some $m$:
$$\frac{1}{n^2}<\frac{m-1}{n}<x<\frac{m}{n}$$
But other than that I'm stucked in the problem.