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I am self learning mathematics and here are some of the tips and techniques I follow while I go through texts like Rudin, Munkres, Artin etc..

I would request the community to mention more techniques/suggestions/advice if they have it in mind.

  • Try to understand the contrapositive of the definition in the book.

Eg:convergence of a sequence $\{x_n\}_{n \geq0}$ in $\mathbb{R}$ is defined as for all $ \epsilon > 0$, there exists $N \in \mathbb{N}$ such that $|x_n-x|< \epsilon$ for all $n \geq N$. The contrapositive would be $\exists \epsilon>0$ such that $\forall N\in\mathbb{N}$ $\exists n\ge N$ such that $|x_n-x|\ge \epsilon.$

  • Try to construct as many examples and counterexamples you can construct. An example sometimes can explain what one page of rigorous explanation can not.

  • While reading theorem and lemmas try to drop conditions and assumptions in the statement. See where the proof went wrong when a certain condition was dropped. This will clearly help in better understanding of the proof.

  • After reading the proof try to summarize the idea in 2-3 lines to check whether you understand the gist or not.

  • Last but not the least solve as many question as you can.

Thanks for reading.

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  • $\begingroup$ I'm not sure of your contrapositive definition of convergence. Perhaps I keep misreading it . . . $\endgroup$ – Shaun Feb 2 at 14:03
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    $\begingroup$ @Shaun Please see the question here math.stackexchange.com/questions/1627002/… I learned it from this post. $\endgroup$ – StammeringMathematician Feb 2 at 14:04
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    $\begingroup$ I see now. Thank you :) $\endgroup$ – Shaun Feb 2 at 14:05
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    $\begingroup$ One thing I find helpful (and fun) is, whenever a new more powerful theorem/ tool is introduced, 'milking' (see this) it to see how many of the weaker theorems I can get. $\endgroup$ – Cardioid_Ass_22 Feb 2 at 14:38
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    $\begingroup$ @spkakkar bhai aaj theek se pada. You made me emotional. Nobody cares for me as much you do :) $\endgroup$ – StammeringMathematician Feb 12 at 10:15

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