I was introduced to real analysis by Johnsonbaugh and Pfaffenberger's Foundations of Mathematical Analysis in my third year of undergrad, and I'd definitely recommend it for a course covering the basics of analysis. I'm not sure if it's still in print that would certainly undermine it as a text!

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I'm using Analysis: With an Introduction to Proof by Steven Lay in my course right now, and from a student's perspective, it's been really good - clear explanations, and a tone of writing that doesn't seem too uptight. I can't speak to other books, but I've enjoyed this one so far! By studying this book, you're gonna be able to achieve an accurate, as well as, an abstract view of concepts like continuity or Riemann-Stieltjes Integral By the way, Mathematical Analysis by Tom M. I personally taught this book once and the result was great.

I'd recommend Analysis Now.

EDIT : Now that the question has been clarified, I'll point out that this is too advanced for a first analysis course. My favourite has always been Introduction to Analysis by Edward Gaughan. I just found out the AMS published the 5th edition. It contains, besides the standard calculus theorems, a very nice introduction to topology of the real line through the study of continuous functions.

I can say that reading this book as a text in my undergrad course largely contributed to myself becoming an analyst. Binmore, Mathematical Analysis. It's not flashy but it's very clean. The proofs are there; they're tidy and I think it's readable. I've used it for this kind of course myself. Might not be a textbook but a very good supplement to a textbook would be the following book Yet Another Introduction to Analysis by Victor Bryant.

Terrance Tao is really an awesome mathematician. The explanations and proof are very clear.

### A Readable Introduction to Real Mathematics

There are many good introductions to real analysis. How about Stromberg's Introduction to Classical Analysis. Heres' another introductory real analysis book: Introductory Real Analysis, by A. Fomin, Dover publications. I recommend Frank Morgan's Real Analysis for its clarity, the concise chapters, and good exercises. It's much more accessible than Rudin I'm not a fan of the Pfaffenberger text. For example, look at the proof of the chain rule. The proof sticks to the "derivative as slope" idea, and so has to consider the special case where one derivative is zero.

This isn't very elegant, and causes confusion in what should be a straightforward proof -- IMO when students are first being exposed to something as elementary as analysis, simplicity should be an overriding concern. Apostol, Buck and Bartle, those are texts that I like pretty well. Or the lecture notes used at the University of Alberta for their honours calculus sequence Math , , , available on-line -- pretty well based on Apostol.

There's a few subtle issues going on here. Some departments view analysis as something people learn after they go through a service-level calculus sequence. Some departments treat calculus as part of an analysis sequence -- ie students only see calculus through the eyes of analysis. What book you choose is largely determined by what path your department is comfortable with. It's pretty nice as a 1 semester course for undergrads and has some nice lead ins to other areas where analysis tools are useful.

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## A Readable Introduction to Real Mathematics | Daniel Rosenthal | Springer

I like the following books, and I feel that they are good books for having a strong foundation in analysis. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Text for an introductory Real Analysis course. Ask Question. Asked 9 years, 10 months ago. Active 2 months ago. Viewed 43k times.

In any case, please explain what " level" means. I'm guessing that this is US terminology; those outside your country won't necessarily know what it means.

Voting -1 for lack of clarity. I think this is university-specific. I've hit it with the wiki-hammer. I edited it to add links, and answer the second part of the question. Many of my complaints about some of the other books that I was looking at were not clear in my head until I saw the same topic in Abbott's book and saw how he explains the purpose of the theorems he presents rather than just giving the theorem and proof. I think for an introductory class students will benefit from that exposition.

The review seems to indicate that. Not much sense avoiding calculus as a prerequisite. It's gentle, complete and walks the reader through a careful presentation of calculus containing many steps that are usually omitted or left as an exercise.

It can also be used for an honors calculus course: I've had friends that have used it for that purpose with great success. Spivak is a beautiful book at roughly the same level that'll work just as well. It's an amazingly deep and complete text on normed linear spaces rather then metric or topological spaces and focuses on WHY things work in analysis as they do.

Lastly, for honor students on their way to elite PHD programs, we now have a wonderful alternative to Rudin and I'm shocked no one's mentioned it at this thread yet: Charles Chapman Pugh's Real Mathematical Analysis , which developed out of the author's honors analysis courses at Berkeley. It's terse but written with crystal clarity and with hundreds of well-chosen pictures and hard exercises. Pugh has a real gift that's on display here. I've never seen any author who does this as effectively as Pugh.

## Reading List

The many, many pictures greatly assist him in this task: all of them serve some purpose, none are throwaways just to fill space. Even if it's just to make a joke see the cornball pic in chapter one showing a Dedekind cut,ugh. Students in an "honors calculus" course at the level of math 55 at Harvard real analysis in disguise who do not see a fairly significant portion of point-set topology by the end of the first semester are in my opinion being done a huge disservice.

Any serious college student who approaches analysis for the first time must know what a proof is, having seen it in Euclidean Geometry back in junior middle school. And I personally like Rudin's, read it when I was still in high school and found it clear, to-the-point, and with a good supply of excellent problems.

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Also, one cannot fault an author for giving slick proofs. I for one prefer slick proofs over tedious, drawn-out proofs unless they're the correct conceptual ones. I don't know where Anonymous was trained,but clearly came from a better system then most students come from. Most high schools in America in have trouble graduating students who can READ,let alone know geometry. It's easy to like slick proofs when you're experienced and well-versed in rigor. Most instructors don't remember what it was like struggling with that fundamental change in thinking that proof creates the first time.

Worse,gifted students think anyone that doesn't find it easy is an imbecile. The collapse of the American secondary school system has sadly affected incoming 1st year math students more then any group. We need to adjust the analysis texts accordingly. The students are NOT "dumber" then in previous generations-as a lot of better trained students snark nowadays-they're simply very poorly prepared. Daniel Take a good,careful look at Pugh's book,especially the exercises.