# How to effectively read a mathematical textbook?

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.