Answer
Let $n$ be some even integer. Then by the definition of even, $n=2k$ for some integer $k$. Multiplying both sides by $-1$ and utilizing the properties of the real number system, we see that $-n=-2k=2(-k)$. But $-k$ is also an integer, because $-1$ and $k$ are integers and the integers are closed under multiplication. Thus, $-n$ is even, so the negative of any even integer is even.
Work Step by Step
This is an example of proof by generalization from the generic particular, as described on pages 151 and 152. The idea is to write the theorem requiring proof as a universal implication ("for all [ _ ], if ..., then..."), name (using a variable) a specific but arbitrary element of the domain with the properties specified in the hypothesis, and show that it must have the properties specified in the conclusion. This proof method works because, even though we go through the logic for a specific element of the domain, we never reference any properties of the element that might not apply to other elements of the domain.