Prove or disprove the inequality with a quantifier

Question

$\exists x \in \mathbb{R}, \forall y \in \mathbb{R}, x - y^2 < 0$

Let $x = y^2 - 1$ then $y^2 - y^2 - 1 = -1 < 0$

This proof got marked wrong, why?

Answer 1

The existential quantifier comes first in the statement, meaning that the chosen $x$ must work for all $y$. You defined $x$ in terms of $y$. A correct proof would be: Let $x=-1$. Then, for all $y\in\mathbb{R}$, $x=-1<0\leq y^{2}\implies x-y^{2}<0$.

What you proved is essentially that $$ \forall y\in\mathbb{R},\exists x\in\mathbb{R}:x-y^{2}<0. $$