If $n$ is even, then $n^2$ is even

Claim:

If $n$ is even, then $n^2$ is even.

Proof:

Assume $n$ is even. Then, by definition, there exists an integer $k$ such that:

Then:

Since $2k^2$ is an integer, $n^2$ is divisible by $2$. Therefore, by direct example, $n^2$ is even.

Claim:

If $n^2$ is even, then $n$ is even.

Proof:

Assume $n$ is odd. Then, by definition, there exists integer $k$ such that:

But it has been shown that for odd $n$, $n^2$ is odd.

This contradicts the assumption that $n^2$ is even.

Therefore, by contradiction, $n$ must be even.