[Math] Tips for understanding and effectively studying mathematical analysis

real-analysis

Just like the aim of elementary linear algebra is to understand the relations of linear maps and to convert mappings (matrices) into Jordon canonical forms to simplify their relations to understand them better (at least this is my understanding of the subject), from what I've been learning, mathematical analysis is studied in order to rigorously establish the fundamentals of calculus.

However, while I may (or may not…) understand the objective of the discipline, I am finding significant trouble being able to solve any question (especially evident given my midterms this week), unless they are very basic. I believe this is due to my lack of 1) identifying the methodology to solve the given question, and 2) laying out a proof utilizing said methodology that is mathematically sound and sufficiently rigorous.

Unfortunately, the problem ultimately seems to reside with my lack of understanding. The book I am currently using for my course is Baby Rudin. The included proofs are extremely succinct, leaving a lot to fill between the lines, not helping my case at all.

Are there any supplementary books that I could be looking into to help me study analysis better? What would be some tips for approaching the subject matter, and tackling questions in analysis courses?

Thank you in advance.

Best Answer

Here's a couple of things that helped me out:

  • Have the definitions down cold. Read them over and over, make sure you don't overlook any details. Definitions help you write proofs. Someone ought to be able to wake you up in the middle of the night and ask you to define what a Cauchy sequence is, and you shouldn't even skip a beat.
  • After reading theorems, try to replicate the proofs, but not in the sense that you will memorize it line by line. Take your time and study why sentence $x$ or result $y$ is there, chances are those are very important.
  • Start with a less difficult text. Try "A friendly introduction to Analysis" by W. Kosmala. Not the most rigorous but very readable.
  • Write, write, write. If you can find a book with end of chapter exercises, where you'd be asked to prove things, then try and do most of them. Make sure you get feedback on your writing. You don't want to develop bad habits early on.
  • Study with a buddy. Can't tell you how many times my study partners found mistakes in my proofs/arguments, and myself in theirs. Also, if a proofs is particularly difficult, then between the two, three of you can probably figure it out. Keep the group small.
  • Write what you want to find, state what you know, use what you know to prove the result. Often, just following this simple step will be half the work.

These are some basic things the will get you on your way. Good luck!

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