It was too much of an encyclopedia for me. I stopped Kolmogorov and Fomin's book almost immediately. On the other hand the one thing I didn't quite like was the excessive use of exercises: every two pages some kind of proof is "left to the reader." Sometimes also people that are not undergrads are going to read the book! Moreover this book treats only real numbers, and sometimes you lose the "big picture." I really liked Abbott's approach: he really makes you understand the logic of things, and you never get lost in the proofs. However, I found the book very clear and rigorous, especially the first 7 chapters. Two critiques I have are: there is a general lack of comments (a bit too much "Theorem, Proof") and there are no images. The one I liked most, and I ended up reading entirely, is Rudin's one: I am a PhD student in engineering and I think the level of the book was perfect to me. I read this question a month ago and I decided to go for three of the most suggested books: Abbott "Understanding Analysis", Rudin "Principles of Mathematical Analysis", and Kolmogorov and Fomin "Introductory Real Analysis".
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