Let

(dropping the restrictions on rationality). It's not all that hard to show that there is a rational number r such that

.

To show this, assume the contrary; there is no such rational number. Let N be a positive integer and consider the numbers

where i ranges over the integers. The distance between two consecutive members of this set is

. If no member of the set lies between m and n, then we must have, for some i,

,

so

.

Since our choice of N was arbitrary, this inequality must hold for all N. An equivalent inequality is

.

But we can always find an N such that

,

by Archimede's principle.

This contradiction shows our assumption that no rational lies between m and n is false.