Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Question

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$.

Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$.

Is that correct?

Answer 1

Hint: Use that $$m^2+n^2\geq 2|mn|$$ so $$|mn|\le 0$$