Better yet, what I'm asking is how do you actually write your mathematics?
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed mathematical concepts introduced in my classes and even did a few competitions; I definitely wouldn't say I was a star of the caliber one meets when one ventures out into the bigger ponds, but I thought I was decent and convinced myself I would major in math when I entered college.
That changed after a couple of years when I hit my first Real Analysis class that used Rudin's book; that was the first class, I think, I took that really required more than "expand-a-definition" type proofs and my struggle to find intuition and understanding there impacted my mathematical self-confidence. I eventually switched majors, with a bit of regret.
One thing that got me, I think, was the veritable explosion of superscripts and subscripts that one encounters for the first time in Real Analysis. I'd often find myself struggling to set up the machinery of what I was trying to prove, lost in the notation. How do good mathematicians format their work on paper so as not to get lost in the $i$s, $j$s, and $k$s and keep track of what they're investigating? I remember dealing with subsequences of sequences to show that limits did or did not exist got especially hairy in this way...writing things like $s_{n_{k_{\epsilon}}}$ and remembering what my goal at each "level" was difficult. I'd be interested in knowing if aspiring mathematicians and/or professional mathematicians scribble marginalia or have a system to overcome such problems.
Another thing that got me were what I personally called "consider..." statements. Many times, on this site, the most talented commenters will say "Consider $f(n)$" or "Consider transformation $T:$ $U \rightarrow V$" that in the first case gives a summation that wonderfully telescopes/has an obvious bound, or in the second case transforms the problem into a trivial application of the rank-nullity theorem, or something like that. Mathematics is a subject replete with geniuses , I understand that, but how do mere mortals investigate such functions and "massage" them into doing what they want? When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong? (Aside: I've been given "use numerical examples" as advice before, but sometimes I think to myself, "I've been dealing with $\mathbb{Z}$ since I was 6 years old, and not so much with Dedekind's definition of the real numbers...")
There's lots more I could ask, but I want to keep this question tractable, so I guess I might summarize by asking: How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it? I know there will be no strict formulas anyone can give; mathematicians are scientists of the abstract; I understand that the subject is acclaimed partly because it's so intellectually and individually demanding. But I think even acclaimed scientists draw on Springer's Protocols and Nature Methods...There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level. I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.
(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)