Good First Course in real analysis book for self study

Does anybody know of a good book in real analysis for self study for a beginner? What about Analysis 1 by Terence Tao?

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I think, Rudin's "Principles of Mathematical Analysis" is very nice. I didn't read Tao's book, though. –  Ilya Aug 29 '12 at 7:45
See here, here, and here. –  Michael Greinecker Aug 29 '12 at 7:46
I like Muresan's book very much. It is heavily based on Walter Rudin's texts, but it contains many interesting results. books.google.it/books/about/… –  Siminore Aug 29 '12 at 7:52
reference-request should not be used as a standalone tag; see tag-wiki and meta. –  Martin Sleziak Aug 29 '12 at 10:20
Rudin (I have taught from it myself) is very terse. It is therefore best used with an instructor, rather than for self-instruction. –  GEdgar Aug 29 '12 at 20:39

Having had my first course in real analysis taught from Tao's Analysis I, I can honestly say that, for a beginner, Tao's book is a great resource. Tao start from the absolute beginning, setting up the Peano postulates and then constructing $\mathbb{N},\mathbb{Z},\mathbb{Q}$ and $\mathbb{R}$ one by one. From the Peano postulates one has the natural numbers $\mathbb{N}$, the increment operation and mathematical induction. Addition is then defined in terms of repeated incrementation and the various properties of it are derived. $\mathbb{Z}$ is developed as the completion of $\mathbb{N}$ with respect to equations like $5+a=3$ and $n+b=0$, where $n \in \mathbb{N}$, and it's properties are proved, either in the text or as an exercise. $\mathbb{Q}$ is constructed from $\mathbb{Z}$ as it's completion with respect to division and yet again its properties are developed. Tao then introduces sequences and Cauchy sequences and constructs the real numbers as a completion of $\mathbb{Q}$, though this is a bit more conceptual.

Thus, Tao assumes no real previous knowledge of these number systems and their properties. Tao also develops all the necessary set theory along the way. This really helped me later to understand sequences, series, derivates and integrals and their properties, as I had a good feeling for the number systems we were working over because Tao had built all the machinery up step by step.

Overall, a great introduction.

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+1 I like Tao's books a great deal for exactly this reason:Most programs treat the construction of the real numbers as archaic drudgery best avoided.I think analysts don't realize how important that "getting your hands dirty" construction in developing a deep understanding of the properties of the real numbers and why it CAN'T be otherwise.It also lets students derive and familiarize themselves with basic inequalities in a way an axiomatic approach simply doesn't do. I think it's well worth the spade work to do in class. –  Mathemagician1234 Aug 29 '12 at 23:14

You could try 'Understanding Analysis' by Stephen Abbott. Personally I found it to be a fantastic book - in fact, it is one of my favorite mathematical texts, regardless of topic.

http://www.amazon.com/Understanding-Analysis-Stephen-Abbott/dp/0387950605

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I was going to recommend this book. This was used in my first analysis class. What is great about it is that Abbott weaves a little bit of a narrative into the presentation of the material. The exposition is great. This all said, the book is rather introductory. –  nayrb Aug 29 '12 at 19:06
I am using this book right now.I have not got far, but I like the treatment. –  user37450 Aug 31 '12 at 14:37

I am particularly fond of this set of free lecture notes (virtually verbatim) of Fields Medal winner Vaughan Jones's beginning real analysis course. It is his own treatment: beautifully done, self-contained, and very accessible to first-time students. The proofs are elegant and enhance intuitive understanding. It really functions as a complete text.

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You should try "Foundations of Modern Analysis" by J. Dieudonne.

It is a great book written by a great mathematician so you will be learning from true Master.

In his book Dieudonne do not assume any previus knowledge about analysis or even mathematics at all.

You need only some intelectual maturity to read it.

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In my humble opinion, and I love Dieudonné's Treatise, this might be hard to read, for a beginner. –  Siminore Aug 29 '12 at 11:52
@Siminore of course you are right, this book requires much work from the reader. But if someone would like to learn analysis together with some basic topology and functional analysis the first volume of Dieudonne's treatise might be a great way to get fast into (quite) modern analysis. –  Godot Aug 29 '12 at 12:03
I completely agree about the high quality of Dieudonne's book. Although it was written in 1968 (more or less), it is still an invaluable source for students and also for teachers. –  Siminore Aug 29 '12 at 12:26
There is a similar book written by Schwartz(the creator of distribution theory) who was also a member of Bourbaki. –  Makoto Kato Aug 29 '12 at 12:53

Our professor uses A Friendly Introduction To Analysis. I think it reads fairly easy. http://www.amazon.com/Friendly-Introduction-Analysis-Witold-Kosmala/dp/0130457965

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