I'm working on a basic proof for my intro proof course. The text is Analysis with an Introduction to Proof by Lay. This question comes from section 2. I am asked to proof or disprove the following:
If $x>5$, then there exists a $y>0$ such that $x^2>25 + y$
So my work follows:
Proof: Suppose $x$ is an arbitrary real number greater than 5 and choose $y=x-5$, then
$25 + y = 25 + (x-5)$
Which is less than $x^2$ when $x>5$. Therefor for all $x>5$ there exists a $y$ namely, $y=x-5$ such that $x^2>y+25$
So that's what I have. At this point I'm not entirely convinced that I've shown enough to proove my statement. I'm still very green when it comes to writing proofs and I feel as if my conculsion is not obvious. I think I'm on the right track here but I could use some direction. Also I've been scolded more than once for doing proofs backwards, I'm hoping that I've alleviated this in this problem but again, this is all still pretty new. Any help is appreciated.