5

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.

Ben Grossmann
  • 225,327
Shweta Aggrawal
  • 5,501
  • 2
  • 15
  • 48
  • I'm not sure of your contrapositive definition of convergence. Perhaps I keep misreading it . . . – Shaun Feb 02 '19 at 14:03
  • 1
    @Shaun Please see the question here https://math.stackexchange.com/questions/1627002/contrapositive-convergence-of-a-sequence I learned it from this post. – Shweta Aggrawal Feb 02 '19 at 14:04
  • 1
    I see now. Thank you :) – Shaun Feb 02 '19 at 14:05
  • 1
    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. – CardioidAss22 Feb 02 '19 at 14:38
  • When you read a lemma/theorem, it becomes another tool in your toolbox. So, think of scenarios, where it can be applied to obtain some new information (imagination required). For example, Monotone Convergence Theorem can be used to obtain Nested Intervals Theorem. Bro, have you sold your phone or what ? Your presence is being missed. – spkakkar Feb 06 '19 at 21:19
  • Given a theorem, try to make a simple question that applies this theorem. Then, try to obfuscate it by either adding some tricky information, or by twisting it so that no trace to the theorem being applied remains. This will help you smell the theorems working behind unseen questions in exam. Aaja bhai vaapas, kyu angrejo ke saamne haath judwa rha h... – spkakkar Feb 06 '19 at 21:31
  • Last but not the least, $\textbf{regularly}$ discuss your observations for half an hour with a friend, like me. – spkakkar Feb 06 '19 at 21:45
  • 1
    @spkakkar bhai aaj theek se pada. You made me emotional. Nobody cares for me as much you do :) – Shweta Aggrawal Feb 12 '19 at 10:15

0 Answers0