Should I prove every theorem in the textbook myself?

Question

I'm currently self-teaching analysis by reading From Real to Complex Analysis by R. H. Dyer and D. E. Edmunds. My strategy at the moment is to go through the book slowly, coming up with examples of objects that satisfy each property mentioned, and proving every lemma and theorem (and doing every exercise) myself, before looking at the proof provided in the textbook.

As I go, I'm creating a LaTeX document containing questions that I can look back at to remind myself of all of what I need to remember (e.g I have "state theorem X" and "prove theorem X" as questions for every theorem, so I can regularly remind myself how to prove things) and the answers to those questions, including proofs for every theorem, and my solutions to the exercises, in the document.

It's working so far, and I feel like I'm gaining a really good understanding of the material, but it's incredibly slow. In fact, my solutions document is longer than the section of the textbook I have gone through so far. I have time, as in I'm not in university, so I can take 6 months to finish the first chapter if I want, but I also don't have time, in that I'm busy studying for other things as well. I also would like to get to the more exciting content that is later in the book, but I don't want to feel out of my depth when I get there.

Is this a good strategy for learning maths from a textbook?

Answer 1

This is certainly a good approach to master the content of your book, but you could probably improve speed as follows:

1. Unless you are really a $\LaTeX$ champion, use pencil and notebook for most of your work (you can still use $\LaTeX$ for the really important things).
2. Keep the examples carefully. This is worth a special notebook (and even a $\LaTeX$ file Examples.tex if you wish).
3. Don't duplicate what is already in the book, in particular proofs. You may try to prove the results by yourself, but once you get the idea, your notes should only sketch the proof by giving hints like: "use compacity" or "apply the intermediate value theorem". You should arrive to the point where these hints allow you to recover the entire proof instantaneously.
4. Do the exercises, but not in $\LaTeX$. First to go faster, but also because you may discover later on that your proof can be improved, or that it can be greatly simplified by some result you haven't studied yet.
5. If you have the opportunity to do so, from time to time ask a mathematician to check your proofs - or ask on this site ! Sometimes, you may discover that some correct but pedestrian proof can be improved by a very simple observation. And once you know the trick, you will save a lot of time.