Answer
Negation: "There exists a greatest even integer."
Proof: There is no greatest even integer.
Suppose that there is some greatest even integer $m$. Then for every even integer $k$, $k\leq m$. Now let $n=m+2$. Then $n\gt$m. But this is a contradiction, because $n$ is also an even integer. Hence, our assumption is false, and we conclude that there is no greatest even integer.
Work Step by Step
We are justified in defining $n=m+2$ by the closure of the integers under addition, and we are justified in asserting that $n\gt$m by the order-preserving property of addition. These are, however, technicalities.