I am currently a first year undergraduate majoring in mathematics. I'm taking an introductory analysis course and find it very hard compared to other math couses. I know that the topics covered in the course are really the basics of real analysis, such as properties of $\mathbb{R}$, sequences and series, limits, continuity, Riemann integral, etc. I work much harder in analysis than in other courses such as abstract algebra, and am spending a lot of time to memorize all the theorems and their proofs mentioned in class. However, when it comes to work out a problem in the book or in the assignment on my own, I'm stuck. My guess is that I never learned how to do math rigorously, and I always rely on my intuition, which proved usually accurate in the past.
The textbook we are using is "Introduction to Real Analysis" by Robert Bartle, 3rd ed., but I also downloaded and use some extra analysis notes from a few professors' webpages.
Could you please give me any advice on how to study analysis? I'm now really desperate 🙁
Best Answer
Memorizing proofs doesn’t really do much for you, at least in the long run; instead, you should try to see what makes them tick. First, what is the structure of the argument? What are the main steps, and what are merely details of carrying out those steps? Many proofs at this stage of your studies have just a single main idea, and everything else is details. Secondly, what kinds of details appear over and over? What basic technical tricks keep reappearing? Those are tools that you want to master for your own use.