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.