How do I improve my skills in constructing mathematical proofs?

[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 staring 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

 

[QuantumQuest]

In my opinion, there's no other way to learn constructing correct and preferably elegant proofs than hard work and all the things that this entails. I think that both practicing more proofs and continuing with other mathematical topics till you are at a decent level and be able to do proofs that you previously couldn't, are things which given the right timing and combinations when you apply them, can lead you to a good learning. There's no secret, magical way or shortcut; the secret is in my opinion to not give up. Experience and expertise are things gained along the way provided that you work in a focused and systematic manner and wholeheartedly.