Problem #WSP-000092

Problem

The pigeonhole principle is often called “Dirichlet’s box principle". Dirichlet made good use of this tool to show a fundamental result in Diophantine approximation, now commonly known as the Dirichlet Approximation Theorem. You will now prove it yourself!

Suppose $\alpha$ is any irrational real number and $N\ge 1$ is any positive integer. Show that there is an integer $1\le q\le N$ and an integer $p$ such that $$|q\alpha -p|<\frac{1}{N}.$$