How would one prove $\forall x \in \mathbb{R}$, $\exists y \in \mathbb{R}$, $(x^2-y < 100)$ in predicate logic.

Question

$\forall \ x \in \mathbb{R}$, $\exists \ y \in \mathbb{R}$, $(x^2-y < 100)$
How would one go about proving this?

Should one use a direct proof or proof by contraposition?

How can one prove this for every x?

Answer 1

Hint: for arbitrary $x$, let $y = (x^2 - 100) + 1$.