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So, in more advanced courses and fields in math like undergraduate and more advanced, how does one make a proof? He makes trials of everything he knows about the problem? What are the instructions to make proofs? When someone reads theorems and other proofs should he make preparations to answer the problems he will try to answer?

When i try i mostly fail solving the problems. Especially in courses like introduction to metric spaces or general topology. What should i do to correct this? When i read the answers to the problems the process of proving them confuses me. Could you help in this?

Also, because i have failed in exams sometimes, and i found them difficult, i find difficulty trying to read and solve problems because of fear.

Thank you.

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    $\begingroup$ I'm not sure what you mean by "makes trials of everything he knows about the problem" (emphasis mine). Could you perhaps clarify that? $\endgroup$
    – J W
    Aug 3 at 11:08
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    $\begingroup$ By making trials i mean, trial and error, the main way a mathematician or a scientist has of solving problems i think, $\endgroup$
    – plants
    Aug 3 at 11:16
  • $\begingroup$ Thank you. You can edit your question using the Edit button/link under the question to include the clarification. $\endgroup$
    – J W
    Aug 3 at 11:23
  • $\begingroup$ Get a copy of "How to Solve it" by George Pólya. There is a summary at en.wikipedia.org/wiki/How_to_Solve_It $\endgroup$
    – alephzero
    Aug 3 at 12:57
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There's no simple switch to turn on to rectify this.

  1. You need to work. And keep working. Not trying from fear is not going to get you anywhere. If you have worse skill/capability, you probably need to work even MORE than those with higher skill. And "giving up" or "not doing homework" are the most common reasons for people not progressing in a course...and especially with self study (one of your interests). "Gotta work" may sound trivial but it is not. It's a common failing.

  2. You need to do partial work even on problems that intimidate you. Rewrite the statement. Draw a diagram. Do a trivial manipulation. At least it gets you into the topic a little bit more actively, versus passive looking at it. How you just came here for an answer (versus doing a Google search and sharing the insights) likely reflects a failing you have in your proof work as well. You need to do AND show some scratch. Not just expect a deus ex machina.

  3. Try to find/use books appropriate for your skill level. There is a very bad tendency on the web (SE especially) of people recommending only the toughest books, even for students that ask for an easy one. If you know you are struggling with benchpressing 225#, back off and try 135#. Or 115# (or whatever). You can do the Rudin and Spivak and all that jazz later (if you choose). But be very wary of following the recommendations for the most skilled, advanced (or even already know the topic!) students. There's a reason why Feynman lectures failed at Cal Tech. And why students (even there!) benefited more from a standard approach, at least at first.

  4. Make sure that you have (or at least firm up as you go), manipulation ability and prerequisite course skills. For instance, I was weak at partial fractions (in algebra class), but in the course of integral calculus and ODEs (Laplace transforms), I corrected that issue. But if you come on something like that (e.g. some aspect of set theory, or of series convergence testing), don't dismiss the error as from an earlier course. Look at it as a "free" opportunity to practice, upgrade, or familiarize yourself with that skill also. I would not get so diverted that you are unwilling to try any new material. But at least firm up old things that were not quite mastered.

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    $\begingroup$ I agree with most of this but disagree or at least don't know what to make regarding part of the second point. It seems you're suggesting this question should not even be asked to begin with. Unless by "show some scratch" you mean that they should have added "prior research" to the question, in which case it makes sense, but the way you've phrased it implies exclusivity with even asking the question in the first place. The thing that gets me with this is the "expect a deus ex machina". If I was asking this question - and I would on other topics - I would not be expecting that, but rather $\endgroup$ Aug 3 at 9:25
  • $\begingroup$ would be expecting a series of general guidelines to lay out a broad roadmap. Effectively what you've already laid out here, without this kind of chidey bit that effectively penalizes simply asking. So you cannot just necessarily assume that this poster would be thinking that. People can be thinking different things than you think they do. Though, I'm not really comprehending this bit anyways so what I'm saying might be totally whack. $\endgroup$ Aug 3 at 9:26
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Another text on this...

How to Read and Do Proofs, Daniel Solow
picture

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Perhaps Beck and Geoghegan's "The art of proof", Martin's "An Introduction to Proof and Mathematical Vernacular" or Hammack's "The Book of Proof" are relevant. All are free PDF downloads, you can peruse them and see if they are a match for you.

Due to it's placement in most computer science curricula, most introductory books on discrete mathematics include a hefty introduction to proofs. You'll also find loads of lecture notes for such classes around.

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OP: "how does one make a proof?"

This (or its equivalent) could help:

Velleman, Daniel J. How to prove it: A structured approach. Cambridge University Press, 2019. 2nd Edition. Cambridge link

     cover

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