For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with wife and kids, taking more college classes is purely out of the question.

I have a solid foundation in number theory/combinatorics/graph theory, and basic algebra and calculus. Is there a solid lesson plan to take someone from number theory/comp sci to real analysis without a college course?

The winning answer will include:

  1. Textbook suggestion with justification. (perfect answer would be one written for a similar background)
  2. Three common pain points that empirically tend to require instructor intervention (perfect answer would be roadblocks comp sci students tend to have)
  3. list of your own length of theorems that make the transition from number-theory simpler.

This does seem fairly broad, but when I was reading on the meta site, it seems that education affords a certain expected level of subjectivity. However I don't know at all where else I can go to ask questions to a large group of educators. I have tried to narrow down the focus, but I'll understand if the question gets canned.


J.W. asked a really good question in the comments, enough I'll quote it:

Do you mind if I ask what your motivation is for studying real analysis? Would you like to study it for its own sake or do you see it as a stepping stone to another topic (or both)? Where you're trying to get is arguably as important as where you've come from.

My motivation is simple: I want to know how calculus works. I tend to be a rather annoying student. For example, when taking college algebra, it wasn't enough to see the term x^2 in an exercise and just do operations on it. I needed to know why x^2 was x^2. Both College Algebra and Calculus more or less introduced me to all this stuff that "works by fiat." Well, Discrete Math/Combinatorics taught me how algebra works. (Not all the way of course, but it felt far less like magic!) Calculus still appears to be magic to me. It isn't enough for me to know that smart people proved it works... so analysis it is! After I climb this mountain I know there will be other peaks but for now I'll limit myself to just understanding how calculus works. I will likely study Abstract Algebra after this so I can deeply understand hyperelliptic curves. But its also possible that the study of analysis will provoke me to study a deeper topic within analysis too... I'm keeping my options open.

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    $\begingroup$ Welcome to the site! This is an interesting question. Could you elaborate on your math background a bit. What does "solid foundation in number theory/combinatorics/graph theory" mean? For example if you could name books whose content you are familiar with this could help a lot in giving you a good answer. $\endgroup$
    – quid
    Apr 4, 2014 at 22:28
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    $\begingroup$ I used Tucker's "Applied Combinatorics" with external reference Grimaldi's "discrete and combinatorial mathematics" For number theory, I've used "Elementary Number Theory" from Burton, the class used Burton's book by the same title. Also used Jones & Jones. (Really didn't care for Rosen's book.) Graph theory was Yellen. $\endgroup$
    – avgvstvs
    Apr 4, 2014 at 22:56
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    $\begingroup$ I remember several "how do I learn ... by myself" on MSE. Look for courses e.g. on Coursera, trawl the web for lecture notes. When you get stuck, go for MSE. $\endgroup$
    – vonbrand
    Apr 4, 2014 at 23:35
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    $\begingroup$ @vonbrand, There are no courses on analysis on Coursera. Though you did lead me to an algorithms course from Sedgewick, who has an awesome textbook on algorithm design! $\endgroup$
    – avgvstvs
    Apr 5, 2014 at 2:55
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    $\begingroup$ For what it's worth, J. M. Cargill writes here: "When you get into a new area, there is something to be said for starting with the most elementary works. For example, even if you have a Ph.D. in physics, if you are trying to learn number theory but have no knowledge of the subject go ahead and start with the most elementary texts available. You are likely to find that you will penetrate the deeper works more ably than if you had started off with deeper works." $\endgroup$
    – J W
    Apr 5, 2014 at 4:55

4 Answers 4


Some suggestions

Michael Spivak's Calculus

This is really an honors level calculus text, but it might be useful to have around. It's pretty expensive, however.

Stephen Abbott's Understanding Analysis

Back in January 2003, in sci.math, I wrote: I think it's the best written introductory analysis book that's appeared in the past couple of decades.

David M. Bressoud's A Radical Approach to Real Analysis

Takes a more historical approach than any other standard text currently in use. Heavy focus on pointwise and uniform convergence of sequences of functions (power series, Fourier series).

Victor Bryant's Yet Another Introduction to Analysis

All exercises have solutions. Full of pictures, diagrams, and other pedagogical aids. Excellent as a direct sequel to the standard calculus sequence and the book is carefully written with this in mind.


I've been thinking about your specific concerns and I cannot think of anything inherently unique about your situation or your background that would suggest a specific approach. My feeling is that typical differences among students in general would be first order effects, while issues facing someone coming from a computer science background would be second order effects that get drowned out (at the population level) by the first order effects. In fact, your background in number theory and combinatorics probably make you better prepared for real analysis than many (most?) students are after the standard calculus, ODE, and linear algebra courses taken in the first two years of college.

What I suggest is that you visit a university library, bringing a pen and paper to write on, and spend two or three hours browsing the shelves. Real analysis texts will be mostly located in the vicinity of QA 300 to QA 330 Library of Congress classification. Read the prefaces of the texts on the shelves, and perhaps even photocopy the prefaces along with the tables of contents. Jot down titles of books that look interesting, either for primary study or for secondary reading. Make a note of how frequently the book was checked out or how worn out from reader use the book seems. Then do some online searching to see what others say about the various books. Keep in mind that a book no one talks about could still be very good. Most people are only aware of the books they used and perhaps a few other books, and most people commenting online probably have little knowledge of anything that hasn't been widely used in the last 30 years. For example, I think John F. Randolph's Basic Real And Abstract Analysis is one of the best beginning graduate level real analysis texts I know of, but because it appeared in 1968 and because it is an awkward fit for most mathematics programs (too advanced for most undergraduate courses, too leisurely for most graduate courses; in this respect it is a bit like Neal L. Carothers' text Real Analysis), there is virtually no discussion of Randolph's book anywhere online.

A good time to visit such a library is a week or two after final exams, because a lot of books that had been checked out will be returned and the library staff will have had time to reshelve the books.

Early this morning I went through a lot of papers I have related to the teaching of real analysis and made a list of those that could be of use to you if you're especially interested in discussions about students' difficulties in real analysis and people's thoughts on how to minimize these difficulties. I am not including papers that would be mainly of interest to those whose primary focus is on mathematics education rather than on mathematics, because despite this being a mathematics education group, your interest is in mathematics. [The difference for me is whether the paper appeared to be primarily about mathematical ideas or to be primarily about psychological and sociological aspects of learning mathematical ideas.]


[1] Robert Lee Brabenec, Resources for the Study of Real Analysis, Classroom Resource Materials, Mathematical Association of America, 2004, xii + 231 pages.

[2] Andrew Michael Bruckner, Judith Brostoff Bruckner, and Brian Sheriff Thomson, Elementary Real Analysis, Prentice-Hall, 2001, xvi + 677 + 58 pages.

The final 58 pages consist of appendixes: (A) Background (B) Hints for Selected Exercises (C) Subject Index. Like Bressoud's book above, this book does not suffer from the problem pointed out by Bressoud in his review of another book below. Also, it's now freely available on the internet.

[3] Kenneth Ownsworth May (editor), Lectures on Calculus, Holden-Day, 1967, vii + 180 pages.

Contains some very interesting selections, written by 9 different people. I encourage looking at this if/when you visit a university library to browse real analysis books. Some (but not all) of the chapters will probably be too advanced for a beginning real analysis student.

[4] Joanne Erdman Snow and Kirk Edward Weller, Exploratory Examples for Real Analysis, Classroom Resource Materials, Mathematical Association of America, 2003, xv + 141 pages.

[5] Robert Michael Young, Excursions in Calculus. An Interplay of the Continuous and the Discrete, Dolciani Mathematical Expositions #13, Mathematical Association of America, 1992, xiv + 417 pages.

This book is an extremely well-cited encyclopedic exposition with 463 references, one of which is Young's book itself. In addition, on pp. 397--408 a reference citation is given for each of the book's 417 problems (or the problem is stated as currently unsolved).


[6] Teaching Mathematical Analysis to Students, Teaching and Learning Undergraduate Mathematics (TALUM) archive.

Articles written by members of the TALUM content working group: Johnston Anderson, Keith Austin, Janet Jagger, Frank Jellet, David Tall.

[7] David Marius Bressoud, Review of Undergraduate Analysis: Second Edition by Serge A. Lang, Mathematical Intelligencer 20 #1 (Winter 1998), 76-77.

From Bressoud's review "Throughout this book, all of the motivations and the intuitions and the connections have been wrung out of the mathematics." The 2nd half of Bressoud's review discusses his experience with teaching a second semester undergraduate course in real analysis that made heavy (but not exclusive) use of Thomas Hawkins's book Lebesgue's Theory of Integration. Its Origins and Development.

[8] Thomas Arthur Alan Broadbent and others, The first encounter with a limit, Mathematical Gazette 19 #233 (May 1935), 109-123.

This is a written report of an extended discussion on introducing limits to students that occurred at the January 1935 Annual Meeting of the [British] Mathematical Association. Most of the participants were teachers at various university preparatory schools, the exceptions being Broadbent (at University of Reading) and brief appearances by Godfrey Harold Hardy and George Neville Watson. See also the letters to the editor in the same issue of this journal by F. C. Boon (on pp. 131-132) and N. M. Gibbins (on pp. 132-134), as well as Dockeray (1935) below.

[9] N. R. C. Dockeray, The teaching of mathematical analysis in schools, Mathematical Gazette 19 #236 (December 1935), 321-336.

This is a written version of a paper given on 23 February 1935 at a meeting of the London Branch of the [British] Mathematical Association. Immediately following this article, on pp. 336-340, is a written report of the discussion at the meeting that followed the talk. Related to this is Broadbent (1935) above.

[10] Marianne Freundlich, Eventually, Notices of the American Mathematical Society 45 #5 (May 1998), 599-600.

Personal reflections about the author's struggles to understand epsilon-delta definitions and proofs in analysis.

[11] Gurcharan Singh Gill, A survey of texts for a one year course in real analysis at the advanced undergraduate or beginning graduate level, American Mathematical Monthly 75 #9 (November 1968), 1033-1035.

[12] E. J. Moulton, The content of a second course in calculus, American Mathematical Monthly 25 #10 (December 1918), 429-434.

Dated, but contains useful observations about the teaching of certain topics in real analysis. Note the follow-up response: David Clinton Gillespie, Advanced calculus or differential equations, American Mathematical Monthly 26 #5 (May 1919), 189-190.

[13] Edgar G. Phillips, On the teaching of analysis, Mathematical Gazette 14 #214 (December 1929), 571-573.

[14] Kenneth Allen Ross, Genesis of elementary analysis courses, Notices of the American Mathematical Society 60 #9 (October 2013), 1179-1180.

[15] Henry Scheffé, At what level of rigor should advanced calculus for undergraduates be taught?, American Mathematical Monthly 47 #9 (November 1940), 635-640.

[16] David Bernard Scott, On teaching analysis, Mathematics Teaching 42 (Spring 1968), 34-41.

[17] Rolph Ludwig Edward Schwarzenberger and David Orme Tall, Conflicts in the learning of real numbers and limits, Mathematics Teaching 82 (March 1978), 44-49.

[18] David Bernard Scott, On teaching analysis, Mathematics Teaching 42 (Spring 1968), 34-41.

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    $\begingroup$ Yes, I realize this. I also only commented in another thread where I probably could say something of interest, but I'm a bit "posting fatigued" after this two-part math overflow answer: Part 1 and Part 2. $\endgroup$ Apr 7, 2014 at 21:36
  • $\begingroup$ @DaveLRenfro: Should you find the time and energy, I look forward to reading your thoughts on roadblocks and major theorems. $\endgroup$
    – J W
    Apr 8, 2014 at 16:24
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    $\begingroup$ @J W: I have a large folder of papers on issues related to teaching real analysis. It's at home, but I'll look at it after work today and bring it with me tomorrow. Looking over those papers will help me formulate some specific issues to bring up. $\endgroup$ Apr 8, 2014 at 17:22
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    $\begingroup$ Thank you for taking the trouble to add such an extensive amount of material, including numerous references, to your answer. I would upvote again if possible. $\endgroup$
    – J W
    Apr 10, 2014 at 7:08
  • $\begingroup$ @DaveLRenfro +1 extremely nice answer. If it's not too much trouble, could I ask if you have any suggestions for a good text in numerical linear algebra, numerical analysis, or abstract algebra? I particularly enjoy the books which take a historical approach, such as Bressoud's book which you have mentioned; but of course I would welcome any good text. Thanks a lot! $\endgroup$
    – Ovi
    Sep 1, 2019 at 21:43

I think the main advantage of your background is that you have had exposure to proofs and concepts such as sets, functions and relations. Being familiar with this "Chapter Zero" material will help you get started with analysis, but keep in mind J. M. Cargal's advice to start with an elementary text.

Cargal's website lists numerous real analysis texts, although I would add Real Mathematical Analysis by Charles Pugh, possibly as a second text. See also MSE posts such as this one and this one.

I suggest trying at least a couple of texts and seeing what works best for you.


Initial difficulties in learning analysis include seeing the need to prove "obvious" results from calculus. A good supply of counterexamples can help, e.g. Counterexamples in Analysis (or Counterexamples in Calculus, as Dave Renfro kindly suggested), although too many can be distracting. Related to this is learning to pay careful attention to definitions. You may find Ideas from Mathematics Education by Alcock & Simpson worth looking into, or Alcock's wonderful little book, How to Think About Analysis.

Working with inequalities and finding suitable bounds for various quantities can also be a stumbling block. This is characteristic of analysis.

  • $\begingroup$ SO you've nailed the textbook problem. I'm going to use one of the elementary texts from Cargal's website coupled with the one targeted at computer scientists. Its always helped me to write programs to better understand problems. The only thing missing from my question is: 3 common pain points for learning analysis. This could be items that you wish you knew better when learning it yourself. Or ones your own students tend to trip up on. $\endgroup$
    – avgvstvs
    Apr 6, 2014 at 18:48
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    $\begingroup$ Counterexamples in Analysis is probably too advanced for avgvstvs at this point. I think Sergiy Klymchuk's Counterexamples in Calculus would be a better fit. $\endgroup$ Apr 8, 2014 at 13:52
  • $\begingroup$ @DaveLRenfro: Thanks for pointing out Counterexamples in Calculus, as I was unaware of its existence. $\endgroup$
    – J W
    Apr 8, 2014 at 16:14
  • $\begingroup$ And "Counterexamples in Calculus" is amazingly cheap too! $\endgroup$
    – avgvstvs
    Apr 9, 2014 at 23:25

I'm a bit surprised nobody mentioned Rudin's Principles of Mathematical Analysis. It's a difficult book (mostly because it's very concise for its level), but I think that, since you have experience with mathematical proofs per se, you might be satisfied with it. (Though I would strongly suggest borrowing a copy from a library, studying a chapter or two and only then making the decision to buy it or not.) In fact, its conciseness might even be an advantage - there won't be many distractions along the way to the "how does calculus work" part.

That said, I have some experience of self-teaching (not a lot of maths though, but a foreign language), and I have to say that what worked for me really well was to study the same topics from 2-3 books simultaneously. This would almost certainly be tougher idea with analysis, since different authors may use different (but equivalent) definitions of e.g. Riemann integral (in this case, one man's definition is another man's theorem) - but again, this might as well give you more insight.

  • $\begingroup$ Well, I truthfully own an international edition I found for $9. My only issue is that there's alot of background knowledge I don't have in regards to raw mathematics. Yes, I've been exposed to proofs, but I got shocked in my graph theory course introduced a proof solved by means of a derivative: Computer science doesn't use much calculus--not in pedagogical terms. That's because computers can only approximate real numbers. So there's reason to suspect I'm missing some important background to tackle Rudin. (Hence the comp sci constraint to the problem.) $\endgroup$
    – avgvstvs
    Apr 9, 2014 at 23:02
  • $\begingroup$ That said, you may well be right--we comp sci guys LOVE concise representations, but I wanted to ask the question from as novice a position as possible to get honest answers. :-) $\endgroup$
    – avgvstvs
    Apr 9, 2014 at 23:04
  • $\begingroup$ That probably requires some explanation: I'm aware that the math we use in comp sci, doesn't really go terribly beyond the complexity of the binomial theorem. Most of our work truly relies on the shoulders of the integers, which means that in mathematical terms, a mathematics PHD is at least 300 years ahead of us. $\endgroup$
    – avgvstvs
    Apr 9, 2014 at 23:08

Numbers and Functions (Steps into Analysis) by R. P. Burn.

If you want to learn the basic theory of real analysis, this is the best choice. If you check it out, I think you will agree.

The material is similar to that of Victor Bryant, but with full detail. Similar to Abbott, but more extensive. More basic than Bressoud. Also, it is designed for "self teaching", which is what was asked for.

One of my favorite maths books.


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