Proofs and learning them

gop

Hi

First of all, I would like to mention that I can do proofs that involve algebraic manipulations (in a field i.e.) pretty well,
or proofs that involve epsilon-delta arguments or mathematical induction.
However, at the moment I'm reading "Principles of mathematical analysis" and I have a hard time to do the proofs on my own (maybe I can solve 1/3 of them).

I am used to write down the hypotheses and the conclusion in FOL (logic) and then I try to manipulate it to arrive at the conclusion.
This however (in my experience) works only well for proofs with a reasonably number of quantifiers (i.e. epsilon-delta arguments).

So my question would be how do you start a proof that involve a lot of quantification by writing it down in FOL or do you use another method?


Another type of proofs, like the one why every real number has one unique n-th root, are a completely mystery to me. While I can follow it without problem I can't claim that I would have ever thought of that particular step etc..

What would you recommend? Just to practice more proofs (even if that means starring at the same equation for hours without solving it) or to continue with other mathematical topics and eventually reach a level of experience where the proofs can be done (I did the second in some fields of computer science and it worked quite well; however, this were obviously no proofs but rather examples of how to apply certain technique in a unusual way i.e.)

thx