How to study for analysis?

Question

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 :(

Answer 1

An introductory analysis course is when you find out that you don't quite know what the real line is. The course is supposed to be an important landmark in your route toward becoming a mathematician. It is supposed to be hard. Don't spending a lot of time memorizing theorems: try to understand what they say and how they fit together. For the major theorems, try to understand exactly where the completeness of the reals comes into play. (Those theorems are probably equivalent to completeness!)

Although most of what I've said above apparently applies to any course, analysis is different than say algebra because it is about some of the things you have seen in calculus, especially the real numbers and you think you know those well. Well, you don't. :-)

Answer 2

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.

Answer 3

Try to see why the stipulations in the statements of the theorems are necessarily there and what goes awry if they are not there.

Answer 4

+1 for the question. I find analysis more interesting subject than algebra etc. Looks like you're talking about Real Analysis. To study real analysis, it is very essential to learn about what sets are, and how to differentiate and integrate, how to find limits, how to check its continuity? Before studying real analysis, I read a good book on set theory (SET THEORY AND LOGIC by R. R. Stoll) and made my opinion clear about Set-Theoretic Notation and Terminology. I studied topics in following order:

1. Part I
   • Set Theory and Fundamentals about it
   • Differentiation and Integration
   • Integers, Rational, Natural Numbers
2. Part II
   • Real Numbers, Bounded Sets and Real Sequences
   • Elementary and Real Valued Functions (of single variable)
   • Limit, Continuity and Derivability
   • Riemann Integral
   • Improper Integrals
   • Convergence
3. Part III
   • Real Valued Functions of Several Real Variables: Limit and Continuity
   • Euclidean Spaces (the Set $\mathbb{R}^n$)
   • Partial Derivatives
   • Integration in $\mathbb{R}^2$, $\mathbb{R}^3$
   • Curve Lengths/ Surface Areas

Reading so many books, is not a good way to learn better. Faith in one book and go ahead.

Answer 5

What a beginning student, faced with the difficulties of Real Analysis, needs are examples, tons and tons of well written out and very detailed examples. - ( An example of the theorem is often more illuminating than the theorem itself, at least in the beginning. ) - The best source for examples is really your Analysis professor, but if you are self-studying, or working ahead, this book might help.

https://www.springer.com/gp/book/9783030358433

"Solving Problems in Mathematical Analysis" by Radożycki, Tomasz. I believe this is the first book in a series of three.