Books for starting with analysis I am interested in self-studying real analysis, and I was wondering which textbook I should pick up.
I know all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the exercises).  I have completed a computational calculus course which covered everything up to and including integration by parts (including the substitution method and Riemann sums)
I am currently considering:

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*Principles of Mathematical Analysis by Walter Rudin
From what I have heard this is not very well suited for self-study and that while the exercises are extremely difficult, if you take the time they are worth the effort.

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*Calculus by Michael Spivak
I have heard that while Spivak explains proofs in much more detail than Principles, it doesn't cover all of the material in the latter.

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*Understanding Analysis by Stephen Abbott
I don't know much about this. I have only seen some comments saying that it is an excellent introduction to analysis.

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*Mathematical Analysis by Tom M. Apostol
Extra clarification edit:
I would prefer a book not to ''dumb down'' the material, something that would not hold my hand through every step, something that would force me to fill in the gaps myself instead of explaining every single step. That is why I am currently leaning towards Rudin, but before I decide I would still like some information on the book by Apostol.
 A: Rudin's book is too abstract in some sense because it requires some knowledge or sense of metric topology. Although Rudin explains the basic theory, but I don't think this is not appropriate to beginner.
Spivak's calculus is "calculus". Although it is quite tough, it is not a book for undergraduate analysis.
I recommend three books:


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*Ross, Elementary Analysis: The Theory of Calculus

*Marsden and Hoffman, Elementary Classical Analysis 

*Apostol, Mathematical Analysis 


Ross helps reader to understand one dimensional real analysis. It gives quite good examples and appropriate exercise problem. But the book doesn't cover multivariable things.
Marsden and Hoffman gives a tons of examples and interesting exercise problems. Although it is quite a challenge to reader, the book gives many pictures and good explanation to the subject. Although some part is based on higher dimensional setting, it is quite readable.  I strongly recommend this book. If you read this book, you have to aware that the definition of compactness in this book is `sequential compactness'.
Finally, the book of Apostol gives almost full details to the proof. It covers many topics. Maybe this book is more appropriate to person who want to know more advanced topics.  
I heard that one of my friends says Pugh's Real Mathematical Analysis is good, but I didn't read that book. 
A: I'm saddened I didn't discover this strategy years ago, but have you tried reading MAA reviews?
The MAA has reviewed all 4 textbooks you listed. And you can read reviews of other books.
A: I would wholeheartedly suggest Tao's Analysis I and Analysis II. I think the class for which the book originated as lecture notes used Rudin. Tao's proofs are much less terse than those in Rudin, and he starts from the Peano axioms and builds the real numbers up, which I think is a great way to start a real analysis text. Furthermore, Tao leaves many proofs to the reader, but he does so in an incredibly pedagogical manner.
