Personal experience in finding and writing proofs in general [closed]

As part of mathematical education (even upto high school) one usually encounters proofs of some theorems (like the proof that angles of a triangle add up to two right angles or the fact that there is no rational number whose square is two). However there is not much focus on proofs in high school curriculum so even if one has studied some typical proofs one may not clearly understand the need of proofs and what exactly constitutes a proof.

Essentially a proof is a set of logical deductions from a bunch of agreed axioms / assumptions and hypotheses to the desired conclusion. Finding a proof does require some familiarity with common proofs and proof techniques (for example proof via contradiction). Having said that there may be theorems which have simple statements but reasonably difficult/non-obvious/non-trivial proofs. Such proofs in general have to be studied from books, online sources and for mere mortals (like me) it is difficult to find such proofs all by oneself.

But apart from that most proofs can be found easily if one has studied proofs of some fundamental results in the topic under consideration and one clearly understands the key assumptions and hypotheses and the conclusion to be proved. Often students don't really understand the theorem which needs to be proved and rather try to map some proof of a similar result they studied to the theorem under discussion. Another aspect is that sometimes the typical school curriculum avoids presenting the key axioms/assumptions and instead of proofs of theorems they use the typical infamous line "the proof is beyond the scope of this book/syllabus". Most of the theorems in calculus belong to this category because the typical high school syllabus omits any discussion of the properties of real numbers which form the basis of such proofs.

Next I discuss the issue of writing proofs. The style of presenting a proof is not so much important as its substance / content. Often one learns the style of proofs from the sources from which one has studied typical proofs. In general if one understand clearly how the hypotheses lead to the conclusion then writing the proof is just translating your thoughts into words. While writing a proof one should aim at clarity, unambiguity and must avoid any sort of hand-waving. IMHO hand-waving is an insult to your own intellectual honesty and to the intelligence of the readers. One can start by writing key assumptions and hypotheses with some more details and obvious inferences based on them. If the proof is long enough one can break down it into multiple steps and cover each step separately writing down the conclusion at each step very clearly.

If the proof involves reasonable amount of algebraic manipulation do explain the difficult manipulations using a few words beside the equations. Also one needs to number the equations properly to refer them later and clearly state the assumptions and hypotheses being used. Sometimes while doing so one may find that some hypotheses are not used at all and then one can mention that these hypotheses are redundant and the conclusion holds without them.

When you have written the proof, read it again carefully to see if some part of it can be improved. For example some algebraic manipulation can be simplified or some logical deduction can be performed in a cleaner way. Some part may need a little more elaboration and one might need to reduce symbolism with natural language to enhance readability. Often reading a first draft of a proof leads to many improvements.