You are correct in your assumption that the things I have difficulties are the most basic proofs. I can usually see how things can be proved using contradiction. Like that last one you can assume x is the sup of the set of n and that x-1 couldn't be greater than all n. Then you could show that n+1 or 2 or 3 etc.. is a natural number and that this would be larger than x which would be a contradiction, since it is supposedly the sup.
I'm pretty sure that proof was in your book though :) I would however have no idea how to even approach the continuous function proof. In fact I have no idea what a compact set is.
I can use induction for more difficult induction problems (I've had to use them in cs all the time).
My biggest problem is always starting, which would probably indicate a lack of understanding of ways to use the axioms. And or a not so concrete understanding of the simple theorems.
I'm just going to keep chugging through, and hope that things get better.