Books for starting with analysis

I am interested in self-studying real analysis and I was wondering which textbook I should pick up.

I have knowledge of all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the excercises) and 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:

  • Principles of Mathematical Analysis by Walter Rudin
  • Calculus by Michael Spivak
  • Understanding Analysis by Stephen Abbott
  • Mathematical Analysis by Tom M. Apostol

From what I have heard Principles by Rudin 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.

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

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

Extra clarification edit:

I would prefer a book that would not ”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 make the decision I would still like some information on the book by Apostol or any other options that might be suitable for me.

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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:

  • 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.