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?
 A: 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.
https://sites.google.com/site/math104sp2011/lecture-notes
A: 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.
A: R.G. Bartle: Elements of Real Analysis http://www.amazon.com/Elements-Real-Analysis-Second/dp/047105464X/ref=sr_1_2?s=books&ie=UTF8&qid=1346422736&sr=1-2&keywords=bartle+real+analysis
V.A. Zorich: Mathematical Analysis I, II  http://www.amazon.com/Mathematical-Analysis-II-Universitext-Zorich/dp/3540874534/ref=sr_1_1?s=books&ie=UTF8&qid=1346422912&sr=1-1&keywords=zorich+real+analysis
are two good sources of problems . . . 
A: 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. 
A: 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
A: 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
A: I see there have been quite a few wonderful suggessions but I would like to add mine. Since I have been there, I think I can help you with your dilemma. 
First of all I would suggest to use 
1) Rudin's "Mathematical Analysis" . Most of us think that it is a difficult book to start     with. I agree. But I believe that the problem lies in our minds. It is accessible. You will need time. You have plenty of it now. Each time you struggle with a section, you work on it, you grow and that's how we learn. Solve all the exercises from it. We all learn by solving problems.
else you can follow "Introduction to Real Analysis" by Bartle and Sherbert"
2) Solve the books "Problems in Mathematical Analysis", VOL-1,VOL-2 and VOL-3. These are great resources to learn from. Keep these with you and breathe with them.
