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As an undergrad student of applied mathematics, I have something to say that make's me ashamed of myself. I suck at proving things in mathematics and i know that if I don't get better in doing this right now, the cost will be greater in the future.

I really want to get better at proving things, so I ask you folks can suggest how I can get better at proving: any links, resources?

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    $\begingroup$ I went to math SE and an user said that this question is more suitable in here. $\endgroup$ – Occhima Jul 1 at 20:43
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    $\begingroup$ This is a stumbling block for many. Lots off effort will eventually get you over this hump. It would help if you mention some of the proof oriented courses you've taken so far. Just going back and working on proofs related to those will be helpful. $\endgroup$ – Sue VanHattum Jul 2 at 17:18
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    $\begingroup$ I feel like none of the answers adequately addresses the question. If a student were to ask in person "How do I get better at proofs?" I don't think we'd just point to a book and say "read." I don't mean any disrespect to the answerers. $\endgroup$ – Mark Fantini Jul 3 at 23:32
  • $\begingroup$ Do a lot of them! Try them yourself (no checking-the-book) starting from the easier ones until you get better at it! $\endgroup$ – David Jul 13 at 12:00
  • $\begingroup$ Do you mean that you have trouble translating informal reasoning into a written proof, or you have trouble thinking of the informal reasoning to begin with? Or both? $\endgroup$ – Tanner Swett Jul 22 at 22:48
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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.

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    $\begingroup$ Vellemman's book is comprehensive, thorough and clear. $\endgroup$ – Clive Long Jul 6 at 11:12
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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.

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

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  • 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.

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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.

         

Proofs and Refutations


And I still have much to learn!

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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.

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    $\begingroup$ What is a demand-side proof as opposed to a supply-side proof? $\endgroup$ – Chris Cunningham Jul 20 at 3:27

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