In one direction ( ) we are given: and for proof by contradiction.
In the other direction ( ) we are given: and then do what you said in your earlier post.
I apologize if I am misinterpreting something. All of my comments are directed toward your proof by contradition. I have not addressed any comments to the proof in the other direction.
One direction: Given that and use proof by contradiction to show that (my thinking).
Your other direction looks good.
Okay, so to be clear. You agree that your proof of the direction is incorrect? And now you are working on the proof. And you have decided to try to prove it by contradiction. Is this all correct?
I guess what I am asking is that is it because my proof is awkward that its not correct (i.e. using direct and proof by contradiction)?
Simply because the direct proof is shorter. From what I have seen (I may be wrong) proof by contradiction is usually only used if a direct proof is either lacking or obscenely difficult.