rr00 wrote: How do you come up with a solution or proof easily? I can do simple few line proof. How does one do proof that is couple pages long?
Short answer: mathematics isn't easy.
Longer answer: Sometimes, long proofs are actually just a lot of easy steps. Those ones are not such a big deal - you know exactly what you have to prove, but maybe its like 10 different things you have to check. These are kind of easy "verification" proofs. The harder proofs are when there's just one statement to prove, and not an obvious way to do it - usually, when I produce a long proof of this kind, it's the result of a lot of trial and error. I usually attempted one proof, and found a hole. Then, in attempting to resolve that, I need a lemma or two, so I have to prove those [and coming up with these requires you to have an intuition for what's happening, generally], and then maybe separate out a couple of claims for clarity. It would definitely be rare for me to immediately see how to do it. It usually takes me several days of thinking about it, and many false starts, before producing such a proof.
rr00 wrote: How do you memorize theorems ,definitions and formula easily? They are too logical statement that there is nothing to grasp with, like visualization and etc. I always try to remember it in verbatim, hold one or two for minutes and have to refer a page again later.
First, you do you get used to logical statements. But you don't really learn a theorem by memorizing the statement word for word - you have to ask yourself "What is this really saying?" and figure out the real meaning/intuition to really understand it. It can take a long time to absorb a theorem - you definitely don't read math books at the speed of a novel. This afternoon, I spent several hours reading one page of a book.
rr00 wrote: How do you study for long time? I easily exhausted after 2-3 hour, feeling faint.
It's very hard to be productive for a long time without a break. Generally, I work on a problem (or reading, whatever) until I get tired, and then I walk around, talk to somebody, or whatever, before coming back to it.