In this article, I’ll present my ideas about how to improve mathematics study, which are the forewords of my handwriting notes on Royden’s “Real Analysis”.

From now on, I will practice using a comparison mode to read several math books at the same time, from which I expect the following benefits:

1. prevent falling into mind traps and being stuck for a long time;
2. study and understand math with a board view and from multiple perspectives;
3. unify those common contents which are scattered in different books. (This phenomenon may be due to an factitious adaptation of the math knowledge into limited semesters for utilitarian students.)

In addition to the comparison mode study, I shall also keep it in mind as an alert that

1. do not transmogrify math study into an immaculate typesetting or copying work without vigorous and recondite thoughts;
2. focus on the gist and progress with the main thread instead of caring too much about delicate details;
3. one’s limited energy and passion should be invested on the most difficult parts or even unsolved problems.

With these consideration in mind, the following notes will be taken in a quick and “dirty” way, where there may exist jumps in the logical derivation. After each phase of such study, the notes will be collected into GoodNotes as a source for slide making and article writing.