Dirichlet's approximation theorem

Remarks

A corollary: if $\alpha$ is irrational, then $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ for infinitely many fractions $p / q$. The proof uses the pigeonhole principle. Among the $N + 1$ fractional parts of $0, \alpha, 2\alpha, \dots, N\alpha$, two lie in one of $N + 1$ intervals of length $\frac{1}{N + 1}$. The convergents of the continued fraction of $\alpha$ give such fractions explicitly.