Proving n is even ↔ n² is even

Definition: An integer $n \in \mathbb{N}$ is even is there exists an integer $k \in \mathbb{N}$ such that $2k = n$.

Proof: Let $n$ be an even integer. This implies there must exist an integer $k$ such that $2k = n$. To determine whether or not $n^{2}$ is even, we can simply look at $(2k)^{2} \leftrightarrow 4k^{2}$. The integer $4k^{2}$ must now obviously be even since $2 \cdot 2k^2 = 4k^{2}$. $\blacksquare$.