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.