TITLE: book recommendation for real analysis
QUESTION [7 upvotes]: During the next quarter at uni, I'll be taking a course in real analysis and since I prefer studying with an additional text I thought I'd come here to look for some book recommendations.
My background: I'm comfortable with linear algebra and single variable calculus, but shakey on multivariable calculus. I did have a slight introduction in to geometry.
For linear algebra I found the common recommendation of Hoffman & Kunze to be very helpful (although the abstractness came as a bit of a kicker at first). 
For Single variable calculus I used the book Calculus by Adams and Essex, which I found frustrating to work with, because it often left me guessing at what they were trying to do or why they were doing it. I also felt that they exercised a certain lack of rigour. (Which may be due to the subject which would make it hard to derive from axioms.)
Details: Though it would be a complementary text, I'd prefer one which will still hold value as a work of reference at a later point in my studies. I've heard that Rudin's Principles of mathematical analysis is one of the better textst out there, but also read several comments discouraging Rudin's book as a first introduction to real analysis, hence my quest to gather more information and maybe get some more personalised recommendations.
The course: The mandatory course literature will consist of lecture notes, but 4 texts are suggested as recommended reading.

T.M. Apostol, Mathematical analysis. Addison-Wesley (1974) J.
Dieudonné, Foundations of Modern Analysis. Academic Press (1960) A.
van Rooij, Analyse voor Beginners.Epsilon Uitgaven, no. 6 (2003)  
R.S. Strichartz, The way of analysis (1995)

Thanks in advance!

REPLY [2 votes]: I would recommend Stephen Abbot's Understanding Analysis as this book serves as a great introduction to undergraduate analysis. For a more advanced reference, you can use Real Mathematical Analysis by Charles C Pugh as this is not as terse as most analysis books are but still sufficiently rigorous. One of it's best features is that it uses images to explain difficult concepts and theorems. Hope this helps.