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$\forall x \in \mathbb{R}$, $\exists y \in \mathbb{R}$, $(x^2-y < 100)$ 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?

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Can you use that for any real number $x$ there exists a real number $y$ such that $y>x$? – Sigur Jan 22 '13 at 0:51
What do you mean by $\forall x \in \mathbb{R}, \exists x \in \mathbb{R}$? Should one of the $x$ be a $y$? – Calvin Lin Jan 22 '13 at 1:08
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. – robjohn Jan 22 '13 at 16:55
up vote 4 down vote accepted

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

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Yes, it would be a direct proof. – Nick Thomas Jan 22 '13 at 5:57

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