That is one direction. Can you use it to do the other direction?
I hope that I understand the notation you are using.
If this is incorrect, please tell us what is correct.
For the opposite direction, I have:
Let
Then there exist an element such that
So then
Implies that
Thus complete the proof. I hope this is right.
There is a second part of the problem:
Prove that at least one of and holds, but they don't have to equal.
Now, so that means I would have two cases. But what gives me trouble is that what two different cases would give me those two different result? Would it be that and ?