Prove that if with , then there exists such that .
Proof. (This is about the point in the blog at which I’m just going to use the basic properties of without much comment. As usual, if something is unclear, leave a comment, and I’ll clarify.)
Then, since and (since and using I.3.5, Exercise #4) and so their product is also greater than 0.
Then, by adding ,
So, letting , we have with