How would one prove this 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?

Can you use that for any real number $x$ there exists a real number $y$ such that $y>x$?
What do you mean by $\forall x \in \mathbb{R}, \exists x \in \mathbb{R}$? Should one of the $x$ be a $y$?
Please don't vandalize this post. There is an upvoted (and accepted) answer into which at least one person has put time and effort. This question might be be of use to other users as well.

Nick Thomas · Accepted Answer · 2013-01-22 05:57:49Z

up vote 4 down vote accepted

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

edited Jan 22 at 5:57

answered Jan 22 at 0:51

Nick Thomas
48010

Yes, it would be a direct proof. – Nick Thomas Jan 22 at 5:57

1 Answer