Take in the definition of a Cauchy sequence.
Assume that the don't eventually become constant. That is, for all integers k, for some integer i and some integer j greater than k.
Since the sequence is Cauchy, there is an integer k such that for all n and m greater than k. From our assumption, we can take n = i and m = j. Then . This obviously can't be true, as the difference between two integers must be an integer itself. QED