How to study for analysis?
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.
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. :-)
While this is doubtless too late for the OP, it may help others studying analysis.
Lara Alcock, who does research on how people understand abstract mathematics, has recently written a book, How to Think About Analysis. It addresses the original question by providing helpful advice on how to study introductory real analysis and what the common pitfalls are. While the whole book is full of useful strategies and tips, I was struck by Chapter 4, which includes advice on how to deal with discouragement and stay on top of the workload. The majority of the book addresses the main concepts in many introductory courses: sequences, series, continuity, differentiability, integrability and the reals.
As for relying on intuition, that can be very useful, but never forget that when there's a conflict between an established definition and your intuition, the definition wins. (However, when a mathematical area is in flux, it may be necessary to revise definitions - see Proofs and Refutations by Lakatos.)
+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:
- Part I
- Set Theory and Fundamentals about it
- Differentiation and Integration
- Integers, Rational, Natural Numbers
- 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
- 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.
I am also studying real analysis from scratch. If you take any standard real analysis book it requires fundamentals from Topology. So I followed this route:
First study Schaum's Outline of Theory and Problems of General Topology.
Secondly study Schaum's Outline of Theory and Problems of Real Variables.
I feel that if I am thorough with the solved problems in the above texts(which is also hard but not as hard as following a standard text books related to real analysis), I can understand real analysis easily. The path will be smooth.
My aim is to understand the basics of Wavelets which has tremendous applications in Signal Processing.
Another book I came across recently was:
Numbers and Functions: Steps into Analysis by R.P. Burn. I wish I would know about this book when I started learning real analysis. This book is for a beginner. It has almost all problems solved. It is asking small questions to welcome the reader to the world of real analysis.
Yet another book:
An Introduction to Proof through Real Analysis - Daniel J.Madden and Jason A.Aubrey. This book is very elementary introduction to real analysis - I feel any newbie can understand the concepts in the book.Authors progressively elevates the high school math to real analysis concepts.