Theorem to show $(m+n)^2$ is even implies $m+n$ is even

Question

I understand that

$\begin{align*} (m+n)^2 &= (m+n)(m+n) = m^2+2mn+n^2\\ &\implies m^2+n^2 = (m+n)^2-2mn\\ &\implies (m+n)^2 = 2k+2mn = 2(k+mn)\\ &\implies m+n\text{ is even}? \end{align*}$

I don't understand how the proof exactly or maybe i copied it wrong