Dirichlet's approximation theorem

Statement

Let $\alpha$ be a real number and let $N$ be a positive integer. Then there are integers $p$ and $q$ with $1 \le q \le N$ and $$|q \alpha - p| \le \frac{1}{N + 1}.$$

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.

Explanations

L1 human

Each real number has fractions $p/q$ very close to it, with an error much smaller than $1/q$. The theorem gives the bound $1/q^2$ for infinitely many denominators when $\alpha$ is irrational.