Direct proof

#direct proof#implication#even odd

Statement

Prove directly: if $n$ is an even integer, then $n^2$ is even.

Hint

Write $n=2k$ for some integer $k$ , then compute $n^2$ .

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Hypothesis. $n$ even means $n=2k$ for some $k\in\mathbb{Z}$ .

Computation. $n^2=(2k)^2=4k^2=2(2k^2)$ .

Conclusion. Since $2k^2\in\mathbb{Z}$ , $n^2$ is even.