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I am currently self studying Baby Rudin because I wanted to learn real analysis and introduce myself to proof-writing before I take upper divs. The only places I have already encountered proofs are diff eq and linear algebra and those classes were fairly easy, so I am still new to proofs.

So I was wondering, how would you recommend I take notes on Rudin, or any math textbook in general for that matter? I want to be able to have a clean, organized, ordered set of notes to reference at all times and come back to when I need to. However, I know it is also probably a good thing to try to prove theorems and other things on my own before looking at the proofs in the book, verify proofs from the book, and perform calculations. But this would make my notes cluttered, unorganized, and overall a bad reference for later.

Should I simply copy down every definition, theorem, and proof word for word? Should I not aim to have an clean, ordered set of notes at all and just use up piles of scratch paper as I go through the book?

Basically, for those of you who have read through Baby Rudin, how did you take notes and what would you recommend as the optimal way to do so?

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    $\begingroup$ "A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos. As Halmos said, I'd suggest building up a repository of some key (counter) examples. Then, I'd get the general ideas behind the proofs (draw pictures if possible) and try to recreate some of the proofs. Then, I'd actually learn the concepts deeply via exercises. $\endgroup$ – Alvin Jin Jan 21 '17 at 9:11
  • $\begingroup$ As with any book on mathematics, knowing the definitions and understanding them thoroughly will enable you to answer most exercises easily and keep you sharp with the content. Add to that everything @AlvinJin noted in his comment. I have only read a bit of the start of Rudin before I realised I'd benefit more from a classical approach first. But after I'm done with that I'm going back to Rudin as my second course. This is why I'm leaving this as a comment. The tiny bit of Rudin I have worked through, I've done by providing my own diagrams and constructing simple examples to aid intuition. $\endgroup$ – shredalert Jan 21 '17 at 9:19
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    $\begingroup$ I would start by asking why do you need the notes ? What purpose are they to serve ? If you take extensive notes then its just the same as the text of Rudin itself and you have that already. Maybe they represent the material in your own words, but note that this does not automatically mean you will retain it. Maybe they are just scratch sheets in which case you can throw them away. $\endgroup$ – Rene Schipperus Jan 21 '17 at 11:24

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