How to learn writing proofs? Could you give me some advice and sources, please? I have to learn how to prove, if I want to continue to study science. For now, the fields I am studying are real analysis and linear algebra.

I guess some of the tips are:

1) finding good books - appropriate for my level of knowledge, with a comprehensible style, yet introducing the concepts to the reader by definitions and proofs. I think a good book in this regard is: Calculus in one variable by Tom Apostol. Still, I am not sure if I've found the right one for learning linear algebra. Any suggestions?

2) Get in used with the mathematical language, which for me is very difficult, as I am a scholar coming from the humanities.

3) Learning definitions well and trying to make a mental picture of everything. Trying to put some effort in solving proof-exercises before asking for any help.

4) In my case, as I don't think I am very intuitive in math and as I miss some background, it would be really good to have a tutor. i.e. private lessons - of course, this means spending money but if this is the only way for me for learning better and faster and passing exams...

5) I have also got some books which introduces the reader to proof writing and learning how to prove but I don't have much time for that readings, unluckily, as I am already overwhelmed by homeworks.

Could you please add some pieces of advice as well as some literature I might find of more help, please?

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    $\begingroup$ If you don't have time for reading, then what's the point of asking for literature that you might find helpful? $\endgroup$ – Gerry Myerson Nov 8 '15 at 11:38
  • $\begingroup$ I mean, I think it is better in this moment reading books about real analysis and linear algebra, instead of books about proof tecniques in general. As I said, I've got the last ones too and started to read them but now I realize I have only time for the first ones, for which I am asking advice here, and for the homeworks, which are mandatory. $\endgroup$ – Always learning Nov 8 '15 at 11:51
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    $\begingroup$ @Alwayslearning: Are you a math student in university? :) $\endgroup$ – H. R. Nov 8 '15 at 11:54
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How to read and do proofs by Daniel Solow is pretty helpful. But learning to proofs, like anything in mathematics requires A LOT OF PRACTICE.

  • $\begingroup$ Yes, I have got this book already and liked it. $\endgroup$ – Always learning Nov 8 '15 at 11:52
  • $\begingroup$ do you have a recommendation for linear algebra as well? $\endgroup$ – Always learning Nov 8 '15 at 11:53
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    $\begingroup$ If you're going to learn a subject, make sure you're studying from the books written by the best people in the field. I would recommend Manin's linear algebra. $\endgroup$ – Brandon Thomas Van Over Nov 8 '15 at 12:02
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    $\begingroup$ @Alwayslearning, that sounds like another question... but look for Treil's "Linear Algebra done Wrong" (it's available for free in the 'net). $\endgroup$ – vonbrand Nov 8 '15 at 12:08
  • $\begingroup$ Thank you! But it is not another question! If you read again the body of my post, as well as the keywords, you should get it $\endgroup$ – Always learning Nov 8 '15 at 12:16

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