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How did you improved your proof-writing skills? Do you consider yourself as a good writer?
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Trial followed by ruthless evaluation by professor or peer. Repeat until you finally stumble onto a correct proof, no matter how long or inelegant. Now do it again. Eventually you know how to check your own proofs.

I don’t know how I would have been able to do it without assistance from another human being. My proofs were usually different (longer, messier) from what was in the book.
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It’s great that you’re seeking ways to improve. For me taking a formal Set Theory course taught me how to organize proofs (carefully labeling assumptions and goal) and how to make quick sketch/outline of the proof before writing it formally. The other part that it does take time: seeing improvements take time. I also have lots to work on (several of my proofs were basically circular logic on my exams). If you want to discuss this more, feel free to do.
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I write fiction in addition to doing math and pride myself on my abilities in both media. Both my proofs and my stories are verbose and *highly* detailed. This often frustrates people who know as much as I do on a topic or more, because they view me as being too pedantic. However, novices appreciate the extra detail, and that is all that matters to me. Accessibility is always a priority of mine.


Anyhow, as for YOU, some of the tips I have to give are:

1) Always write the statement(s) of the question/exercise/result/problems you are being asked to prove BEFORE you write out a proof.

2) Doing a proof is actually two different tasks. The first task is for you to find A proof; your second task is to WRITE UP your proof. To that end, do not bother trying to write things up neatly until you have solutions to ALL of your problems (or however many you can complete before the deadline). Once you have your solutions and are confident they are correct, take your time to focus on writing up your solutions neatly and carefully. It is MUCH easier to focus on improving your proof presentation when you only need to worry about how you write up the math.

3) Clearly indicate everywhere and anywhere where you utilize the hypotheses given in the problem.

4) When I write up proofs with computations, I like giving justifications/explanations in parenthesis to make them easier to follow.
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For me, taking a few courses in symbolic logic really helped. In particular, after writing quite a few formal proofs in Fitch notation, I really felt like I gained a much deeper understanding of what it was that I was trying to do whenever I wrote a mathematical proof, and I think it made me both make fewer mistakes, as well as write much more clearly and systematically.
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Latex
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> Do you consider yourself as a good math-writer?

I'd say I'm good enough. I generally receive positive feedback from reviewers regarding my writing style.

> Did you use any method to grow your writing skills up?

Practice, experience, and listening to people's feedback. Most importantly, listening to feedback on early drafts of a text, and applying that feedback in revisions.

> What makes a proof good for you?

The intuition needs to be clear. The large-scale structure needs to be clear. Beyond those points, it depends on what exactly the proof is about and what it's like.

 > What are your main inspirations?

No particular inspirations I can think of.
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Im not really answering your question, but i wouldnt worry too much as long as the proof works. The first time (for me at least) the proof tends to be kinda ugly. Michael penn said that for him solving a problem is like climbing; you try to reach the top, and once there, its easier to see maybe a better way up. For proofs its the same. You solve it however you can and once solved you try and see a better proof

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