|
Theorem (Dirichlet, c. 1840): For any real number  and any integer  , there exist integers  and  such that
 and
 .
Proof: We may assume  . For each integer  in the interval ![$ [1,n]$](http://images.planetmath.org:8080/cache/objects/3740/l2h/img9.png "$ [1,n]$") , write
![$ r_a = a\theta - [a\theta] \in [0,1)$](http://images.planetmath.org:8080/cache/objects/3740/l2h/img10.png "$ r_a = a\theta - [a\theta] \in [0,1)$") , where ![$ [x]$](http://images.planetmath.org:8080/cache/objects/3740/l2h/img11.png "$ [x]$") denotes the greatest integer less than  . Since the  numbers  all lie in the same unit interval, some two of them differ (in absolute value) by at most
 . If 0 or  is in any such pair, then the other element of the pair is one of the  , and we are done. If not, then
 for some distinct  and  . If  we have
 , since each side is in  and the difference between them is an integer. Similarly, if  , we have
 . So, with  or  respectively, we get
where  is 0 or  , and the result follows.
It is clear that we can add the condition
 to the conclusion.
The same statement, but with the weaker conclusion
 , 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  , with the (nominally stronger) conclusion
 .
|
