TITLE: Why does this step work in this proof?
QUESTION [5 upvotes]: I'm trying to learn discrete math and am brushing up on proofs by reading Richard Hammack's Book of Proof. I'm tripped up on this proof... I understand that it's contrapositive, and why contrapositive is the best approach, but I'm not sure why you should multiply both sides of $y-x > 0$ by the positive value $x^2+y^2$. Can anyone explain this to me?

Thanks!

REPLY [2 votes]: Contrapositive is not necessarily the best approach. We have $y(x^2+y^2)\le x(x^2+y^2)$. If $x=y=0$, then clearly $y\le x$. 
If $x$ and $y$ are not both $0$, then $x^2+y^2\gt 0$, and so by division by the positive number $x^2+y^2$, or if you prefer by multiplication by $\frac{1}{x^2+y^2}$, we get $y\le x$.