Discrete Math and about domain of discourse

Question

Determine the truth value and give the reason.

Domain of discourse is $\mathbb R$

$$\forall x,\exists y, xy\ge0$$

Is the answer true?

Answer 1

The statement is true.

Let $x\in\mathbb R$. Take $y=0$. Clearly, $y\in\mathbb R$. Then, we have $xy=0$, and as a result, $xy\ge0$.

Thus, $\forall x\in\mathbb R,\exists y=0\in\mathbb R$ such that $xy\ge0$.

Answer 2

If $x=0$ we are done, else let $y=\dfrac1x$ to get $1\ge0$.