 Dirichlet's approximation theorem

For every irrational number $\alpha$ and positive integer $N$, there are integers $p,q$ with $1\le q\le N$ such thatThe proof applies the pigeonhole principle to the $N+1$ fractional parts of $0,\alpha ,\dots ,N\alpha$.

Dirichlet's approximation theorem is a result in number theory that provides a way to find rational approximations to real numbers.