Proof with quantifiers

Question

$(\forall x)(\exists y)(x+y=0)$ $x$ and $y$ are real numbers The statement reads: for all $x$ there exists some $y$ such that $x+y=0$ is true. My proof is: take $y=-x$

Is this valid? I'm just paranoid that since the proof is so simple that something is wrong.

Answer 1

Yes, you've got the right idea:

Simply expand a tad to say:

Let $x$ be any arbitrary real number. Then put $y = -x,$ and so $\;x + y = x + - x = 0$. Since $x$ was chosen arbitrarily, it holds for all $x\in \mathbb R$.