book recommendation for real analysis 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!
 A: Apostol's book is very good for first learning analysis. However, it leaves out a number of multivariable calculus topics, such as line and surface integrals, and vector analysis. These topics can be found in other books, including the same author's Calculus, Vol. 2.
Overall, Rudin's book has less content than Apostol's and less detailed proofs. The exercises in Rudin's book tend, more often than Apostol's, to require you to come up with ideas that are very different from those in the main text, or to perform more steps in a proof without hints. For some people, this is an advantage of Rudin, and for others a disadvantage.
I would say that Dieudonné's book is probably the best "reference", because it's very formal and systematic. (For example, the first definition given of the derivative is for a mapping between two Banach spaces.) It also discusses important results in the exercises. It is actually the first part of Dieudonné's nine-volume treatise in analysis. Because of its comprehensiveness, it wouldn't be a good first book to learn from for most people, with the exception of someone with very high ability and motivation. 
You could also consider Zorich's two-volume book Mathematical Analysis. Generally, the first volume deals with differential and integral calculus in $\mathbf{R}$ and differential calculus in $\mathbf{R}^n$, and the second volume deals with various advanced topics. However, even the calculus material in the first volume is taught in a relatively advanced way (for example, using lim sup and lim inf to simplify proofs, or open and closed sets). This could be a good book if you want to both start analysis and learn multivariable calculus properly (i.e., with full proofs and difficult exercises).
Based on a very cursory glance at the book by Adams and Essex, I'd say that, compared to rigorous calculus books like those of Apostol and Spivak, it doesn't seem like great preparation for a course in analysis. There is much less theory, and the exercises are easier. So whether you'd be successful starting directly with Apostol's Mathematical Analysis depends a lot on you. If you find that it's difficult going, then you could try using a book like Ross's Elementary Analysis, which is intended for students who have little background with proofs. 
A: I would recommend Stephen Abbott's Understanding Analysis. It may not be the best choice as a stand-alone text, but that is not what you are looking for. It does however a great job in "filling the gaps". In particular, it explains the motivation, why real analysis was developed and why this level of rigour is necessary. A lot of this is done using counterintuitive examples, showing what can go wrong if one is not careful. And the text gives a sense of the history of the subject, which many other text omit to do.
A: 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.
A: If I could suggest two books, I would say a combination of Walter Rudin's Principles of Mathematical Analysis (3rd edition, 1976) and Charles Chapman Pugh's Real Mathematical Analysis (2nd edition, 2015), as they complement each other well.  Pugh has a strong preference for teaching by informal visual intuition.  His writing is also very colloquial by textbook standards, and it successfully conveys the author's enthusiasm for the subject.  Moreover, the author does a good job in explaining the essence of a concept or a proof in a way that helps the student build intuition.  His exercises are also probably the best out there -- it's a huge collection that ranges from routine (no stars) to hard (*) to very hard (**) to questions the author does not know a complete solution of (***).  If you can complete half of the starred questions, you should be more than ready for a graduate real analysis course that starts off with abstract measure theory.
On the other hand, reading "Baby Rudin" is a must for any serious math student.  It's a book that you learn to appreciate more and more as you become more advanced.  Its style is the diametric opposite of Pugh.  There is not a single figure, and in the first edition (1953) he explicitly warns: 
"Often it is convenient to use geometric language when speaking of sets of real and complex numbers.  It should be clearly understood, however, that proofs must not be based on geometric intuition, although the geometric interpretion may be very helpful in suggesting the steps along which a proof might proceed."
Moreover, the text is lean, with very little discussion or commentary.  Instead, the majority of the text occurs under the headings Definition, Example, Theorem, Proof, Corollary (in that order), with the occasional Remark peppered in.  Although I initially found this format to be intolerably dry, this clarity of this style grows on you.  Furthermore, as your knowledge and understanding grows, you will appreciate how much thought the author put into choosing the order in which to present each item.  In short the organization is extremely efficient, as are the proofs themselves, which are rigorous but do not use a character more than needed.  Unfortunately, the proofs are often so clever that it leaves the student bemused and bewildered.  Getting Pugh's take on the same topic is often extremely enlightening. 
Tl;dr.  Read Rudin to develop the stamina for parsing lean mathematical writing (advanced mathematical texts tend to be written in this style!) and to appreciate how pretty his proofs are.  Read Pugh to develop the correct intuition with respect to geometric or topological concepts and to do his excellent exercises. 
