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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:

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.

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.

I don't know much about this. I have only seen some comments saying that it is an excellent introduction to analysis.

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.

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    $\begingroup$ I suggest you look at Francis Su's lectures on youtube. They are excelent. I believe he is using "Baby Rudin" with his class. Rudin is a bit opaque without someone to sometimes explain the proofs. But it is generally considered the authority. $\endgroup$ – Doug M Aug 18 '16 at 1:16
  • $\begingroup$ Rudin would be a very poor choice for self-studying. Spivak's is good as far as it goes. You could always try a slightly more ambitious book and fall back to Spivak if necessary. I would recommend taking a look at Thomson/Bruckner/Bruckner's Elementary Real Analysis. It's a free (legal) download, so you lose nothing by checking it out now. I think Abbott's book is very good, but he leaves a lot of results as exercises, so you might find that annoying (or not). $\endgroup$ – Bungo Aug 18 '16 at 1:23
  • $\begingroup$ Possible duplicate of Good book for self study of a First Course in Real Analysis $\endgroup$ – littleO Aug 18 '16 at 1:24
  • $\begingroup$ I've read Rudin for self-studying, and I would say that it is a very poor choice of words to say that this book is a poor choice. Bottom line is: read the books (at least glance at their interiors) and get your own opinion. Books are a matter of taste more than people would like to admit it to be. But the book is considered authority worldwide for a reason. $\endgroup$ – Aloizio Macedo Aug 18 '16 at 4:28
  • $\begingroup$ If you really wish to do self-study of real-analysis (i.e. not for purpose of complementing your university education and getting a degree) then a very good option is to go for Hardy's "A Course of Pure Mathematics" for elementary stuff (meaning upto integration of continuous functions) followed by Apostol's "Mathematical Analysis" (for Riemann/Lebesgue integral and Fourier series). $\endgroup$ – Paramanand Singh Aug 18 '16 at 8:53
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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.

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  • $\begingroup$ One more comment: I want to suggest a good supplementary for multivariable calculus written by Shurman. $\endgroup$ – Will Kwon Aug 18 '16 at 1:20
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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.

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