We begin by making the assumption that there exists a greatest even integer and like all contradictory proofs we will end up breaking that assumption.

Let $n$ be the greatest even integer. We can add 2 to it and, since $n$ is an integer, $n+2$ is also an integer. We also have that $n+2$ is even because EVEN+EVEN=EVEN. Hence, $n+2$ is an even integer greater than $n$, and so our original assumption that $n$ is the greatest even integer must be untrue. In conclusion, there is no greatest even integer.