I'm solving the following problem from a discrete geometry book (Lectures on Discrete Geometry, Jiri Matousek).
Prove that for $\alpha=\sqrt{2}$ there are only finitely many pairs $m,n\in\mathbb{N}$ with $|\alpha-\frac{m}{n}|<\frac{1}{4n^2}$.
I don't know a lot about number theory, and so the only tools I have are the ones from the book: Minkowski's (first) theorem and the fact that for $\alpha\in(0,1)$ and $N\in\mathbb{N}$ there are natural numbers $m,n$ such that $|\alpha-\frac{m}{n}|<\frac{1}{nN}$. I don't really see a way to approach this problem using them.
I've seen some similar questions on approximations of $\sqrt{2}$, but they seem to deal with more general cases and use results from number theory, but this is not what I'm looking for.
I'd greatly appreciate any hints on how to show this. Thanks!