Prove that if $\alpha \in \mathbb{R}\backslash \mathbb{Q}$, there exists one (actually an infinity) of integers $p$ et $q$ such that $\big|\alpha -\frac{p}{q}\big| \leqslant \frac{1}{q^2}$ and $0<p<q$.
My attempt : I started by considering $\alpha\in [0,1[$ because if not we can just consider $\alpha - \lfloor \alpha \rfloor$ and then add $\lfloor \alpha \rfloor$ to the $p/q$ we find. But then I really don't know how to proceed. $p/q$ must be close to $\alpha$ but I don't see how to make the link with the denominator.
Could someone help me ?