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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 :(

J W
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Ming
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8 Answers8

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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.

Brian M. Scott
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    To add in how to absorb theorems properly: Check conditions. Why are they nessicary? Can you intutively see why without it, some weird example may make the theorems result fail? Even better, can you write out a counter example properly? The best way to remember the conditions of the result is to have spent time finding examples like these. – Ragib Zaman Nov 22 '11 at 11:18
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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. :-)

lhf
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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.)

J W
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+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.

Srivatsan
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Gaurav Tiwari
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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:

  1. First study Schaum's Outline of Theory and Problems of General Topology.

  2. 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.

Another book: Writing Proofs in Analysis by J Kane. This book introduces proof templates for real analysis concepts.

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

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I too am a 1st year student. My suggestion is: always try to draw pictures in analysis problems. At least in the case of $\mathbb{R}$, I found these extremely helpful. So many times, I was stuck with continuity-related problems, but as soon as I drew a figure, I realized the motivation behind the statements.

I also strongly recommend Terrence Tao's book Solving Mathematical Problems: A Personal Perspective to have some instinctive feeling before solving a problem. This book really made me realize HOW TO THINK.

J W
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sauvik
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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.

nilo de roock
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