Theorem (Dirichlet, c. 1840): For any real number $\theta$ and any integer $n\ge 1$ , there exist integers $a$ and $b$ such that $1 \le a \le n$ and $ \arrowvert a\theta-b \arrowvert \le \frac{1}{n+1}$ .

Proof: We may assume $n\ge 2$ . For each integer $a$ in the interval $[1,n]$ , write $r_a = a\theta - [a\theta] \in [0,1)$ , where $[x]$ denotes the greatest integer less than $x$ . Since the $n+2$ numbers $0, r_a, 1$ all lie in the same unit interval, some two of them differ (in absolute value) by at most $\frac{1}{n+1}$ . If $0$ or $1$ is in any such pair, then the other element of the pair is one of the $r_a$ , and we are done. If not, then $0 \le r_k - r_l \le \frac{1}{n+1}$ for some distinct $k$ and $l$ . If $k>l$ we have $r_k - r_l = r_{k-l}$ , since each side is in $[0,1)$ and the difference between them is an integer. Similarly, if $k<l$ , we have $1-(r_k - r_l) = r_{l-k}$ . So, with $a=k-l$ or $a=l-k$ respectively, we get $$ \arrowvert r_a - c \arrowvert \le \frac{1}{n+1} $$ where $c$ is $0$ or $1$ , and the result follows.

It is clear that we can add the condition $\gcd(a,b)=1$ to the conclusion.

The same statement, but with the weaker conclusion $ \arrowvert a\theta-b \arrowvert < \frac{1}{n}$ , admits a slightly shorter proof, and is sometimes also referred to as the Dirichlet approximation theorem. (It was that shorter proof which made the ``pigeonhole principle'' famous.) Also, the theorem is sometimes restricted to irrational values of $\theta$ , with the (nominally stronger) conclusion $\arrowvert a\theta-b \arrowvert < \frac{1}{n+1}$ .