We have been asked to prove that given an irrational $\alpha$ there exist infinitely many relatively prime pairs $(p,q)$ such that $|p/q-\alpha|<1/q^2$. I have so far found a constructive solution with continued fractions, but only existence is required and I feel like there must be a much more concise and precise solution that does not require introducing continued fractions. As a tip we are told to use the pigeonhole principle.
Any other tips or piece of advice will be helpful. Thank you.
