How to get better at proofs

A great place to start learning how to do proofs is, IMHO, the theorem on friends and strangers.
The canonical proof to that theorem can be considered to be the ‘ideal’ proof in all of mathematics in the following sense: it is the proof of something demand-side (rather than, as is most instructional material, supply-side), that is, it is a proof of something interesting (even counterintuitive), it is accessible to beginners (can almost be given verbally during an elevator trip between floors), it does not rely on potentially inaccurate/misleading geometric figures, and it does not require learning any algebra. Here is the link to the Wikipedia article on this theorem: https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers#:~:text=The%20theorem%20says%3A,are%20(pairwise)%20mutual%20acquaintances.

Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience.

Now, I'm not entirely sure what is the best way to arrange this outside of class. Get a "proof pen pal" with whom you swap proofs and give feedback for how clear it was? An online study-group type situation? Post attempted proofs to SE Mathematics and ask for feedback/improvements? (Kind of like how SE Software Engineering, I think, accepts code-review posts).

When you do that, bear in mind that's how actual professional articles get written. Someone writes a draft of one or more sections, they get sent around to colleagues for commentary, and the draft gets re-written and re-written until everyone gives the thumbs-up to it. So if you can get that process going, then you'll have an additional leg up on the people-skills that a math researcher needs.

I learned more from Lakatos than any other source about what constitutes a proof, and:

• the roller-coaster ride adjusting definitions to clarify the proof claim,
• perhaps realizing that a hidden lemma has a counterexample,
• the need to reformulate the proof statement in light of these ups-and-downs,
• and on and on.

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Proofs and Refutations

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And I still have much to learn!

• As a general introduction, the book Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi can be useful. For example, how to prove that a sequence converges? A detailed explanation is given on pages 168-170.

• As a general rule, to get better we have to do a lot of proofs. A lot of sentences to be proved can be found in problem books on the subject you want to get better. For example, this for linear algebra and this for real analysis.

A great start to doing proofs is working through Daniel Velleman's How to Prove it: A Structured Approach, 2nd Edition.. I've used it many times in teaching, usually as a supplementary text.

Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.