I'll add a little to my post to you. Last night, in bed, lights out, no paper or pencil, nearly asleep, I had an idea. Here goes: There's the law of large nu

I'll add a little to my post to you. Last night, in bed, lights out, no paper or pencil, nearly asleep, I had an idea. Here goes: There's the law of large numbers, the weak version and the strong version. Good authors for the details are M. Loeve, J. Neveu, L. Breiman. I've been through lots of proofs. Likely the easiest proof of the strong law is from the martingale convergence theorem. It's so easy, just f'get about proving the strong law and concentrate on martingales! But a few years ago I noticed that Breiman advised students to think of a simpler case. In a hurry, I didn't. Then, yup, last night, trying to go to sleep, I thought: The simple case is coin flipping, say, 0 and 1. Now flip a coin, say, 50 times. Get all the possible outcomes, that is, sequences of 1's and 0's. Now count the sequences with about 50%/50% 0s and 1s. Before getting out the little formulas on combinations and permutations, or just rederiving them, guess that the fraction of the sequences with 50/50 or near that start to dominate in the fraction of all the sequences. Do this for 1000 flips, 1 million flips, etc. Guess before looking into the details and just before going to sleep, if check the details will find that a the number of flips increase, just as the law of large numbers posits, will find that the 50-50 case so dominates that the rest, as fractions, as probabilities, head for zero. Presto, bingo, convergence! That's just a guess. I still haven't worked out the details. With the details, the guess might be right or might be wrong. But that's a little of how intuitive guessing goes. My guess now is that for guessing that guess was pretty good. I've published theorems that started with guesses less good looking than that one! Lesson: For proving theorems, and finding new ones, intuitive guessing, sometimes starting with simple cases, at times can work.