You start by writing your own examples. Remove all the abstract notation and expand out everything, distribute everything back in so you just have a series to b

You start by writing your own examples. Remove all the abstract notation and expand out everything, distribute everything back in so you just have a series to be summed or integrated. Stare at it for k = 0, 1, 2, etc., like 5 choose 4, 5 choose 3...maybe you can prove this yourself. This is actually how you learn, trying to do it yourself and thinking for a few hours/days/weeks about this problem why the identity is true, and then getting all those 'How to prove it' style books filled with established techniques to see if your identity can be proved by them. Try Terence Tao's book Analysis I https://learnaifromscratch.github.io/math.html https://learnaifromscratch.github.io/math.html you pick up the gist of this kind of math doing basic proofs of natural numbers. You see and do enough of these that you will get the first proof here, where the n+1 comes from, and how you can distribute things in/out of any summation notation. The second integral proof from math stackexchange the same applies, remove all the abstract notation and just write down a few series, distribute in the 1/2pi(i). Do this for a few examples and you can probably figure it out yourself. In the above link there is resources for olympiad and putnam problem solving on YouTube, if you watch some of those you'll see how they train competitors to solve something like this problem. List every tiny idea or strategy you have to prove that binomial identity and pick one that seems the easiest, start there. Fill the variables with corner cases what happens. If it's geometry what happens if you 'stretch and pull' the geometric interpretation by some constant factor. Look at Pascals Array since this is a binomial identity. This is what they do in those seminars if you're interested and it's free for anybody to watch. When all those proof books throw 'implies' and other logic at you, read through this: https://ncatlab.org/nlab/files/MartinLofOnTheMeaning96.pdf https://ncatlab.org/nlab/files/MartinLofOnTheMeaning96.pdf