Teaching yourself how to prove

[reactor]

A fellow TM member asked my help on learning to do proofs (as if I'm an expert or even someone good in proofs!) Anyway, I decided to make my reply a thread. I intent to update/refine/correct this post.

I'm a grad student in math (until they kick me out) with an undergrad in economics. No formal math courses in my undergrad; not even mathematical economics. Needless to say I had to study a lot of mathematics on my own; many times in my free time (free-from-economics time, lol!) and many summers. As far as I remember I improved my proving skills by attempting lots and lots of proofs. To my experience exposure works. Math is probably the most involved field; little reading and lots of thinking; you have to do lots of work outside the class, on your own.

First step: Don't look the solution!
There is hidden value on not looking the solution until you have really thought the problem; some times for a couple of days. Even when you are doing other stuff you a part of your brain wil be considering the proof. Giving up and looking at the solution before you become desperate will not provide you with "experience", "illumination" or "knowledge" but with just a quick relief.
Having said that there is lots of value in studying the proof; reading carefully and trying to reproduce it. You have to reach the level of knowing where each assumption is used in the proof and how the proof would fail without this assumption.

Second step: math-translation.
Reading the proposition extra-carefully and writing down what it "assumes" and what it "asks", really help. For example it might say "the sequence converges to a" and then I have to right down that "for every epsilon >0 there exists N such that n>N implies d(x,a)<epsilon". Re-wording the problem, making the assumptions and assertions "viewable" in mathematical terms is at least 1/3rd of the job.

Third step: your opinion.
The next step is to read again the proposition and think why it should be right or why it should be wrong; this step builds intuition which is more than vital. Having an idea of why the statement is true/false helps you gather the tools of constructing a proof.

Fourth step: your knowledge/ideas
Now you have to think/write down what you know about things involved in the statement. Definitions, relevant theorems etc. These are the building blocks of the proof.

I hope you enjoyed. To be continued...

Recommendation by Macrotime: "How to Read and Do Proofs by Daniel Solow", it is an excelent book. It will give you the tools, and following the advice of reactor you'll surely learn how to do proofs.

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