One thing that should be understood is that this book is used in many university advanced calculus courses. Advanced calculus courses, in (almost) all universities, have very high dropout rates, regardless of the textbook used. This is because advanced calculus is a tough course. Often students take a very soft (i.e., not very rigorous) introduction to mathematical proofs before advanced calculus. However, one must be able to read and write proofs very well before taking AC. Often students are not prepared, fail the course miserably, and blame the book, their teachers accent, or the professor of their "intro to proofs" class, and so on. Before taking advanced calculus I had worked through Spivak's Calculus and read other rigorous math books before taking this course, thus I could write proofs reasonably well and had few problems with this book, and neither will you if your proofing skills are in good shape.
That being said, the book is not bad, at all. It gets the job done. As far as rigor goes, very few things are assumed and almost everything that is used throughout the book is proved. Some of the few things that are not proved but are used throughout out the text are the continuity of functions such as sin(x) or e^x. The problems in the book are generally easy and not very interesting. I mean to say they do not illustrate the "beauty" of mathematical analysis. The amount proofs I found in the book to be "difficult to understand" I could count on one hand. When I encountered such a proof following along with pen and paper working out the details and filling in the gaps on my own would clear things up.
Rigor tends to be a common complaint among the other reviewers. Rigor is generally refers to the development of a theory, by essentially proving every little detail. Generally this means that nothing goes up proven; even if an assertion may be completely obvious, it is still formally proved. Not once in this book have I encounter a proof that lacked in rigor or used a certain technique or result that had not been discussed earlier in the book. Some have complained that the proofs are too long or too short, however compared to other analysis texts that I have seen, the proofs in this book seem very standard.
If your analysis class requires you to use this text, don't think you are doomed or will never be a successful mathematics student. If you can read and write a proof you will do fine, but I would not recommend this book for self study because there are much better, well established, classics in the field of Elementary Real Analysis. Such as "Calculus", by Michael Spivak, "Real Mathematical Analysis" by Charles C. Pugh, and finally, slightly dated but arguably the best, "Principles of Mathematical Analysis" by Walter Rudin.
Edit: If one is willing, going to the "All Comments" section and viewing the two comments above mine proves my point of students struggle with an Advanced Calculus course and take it out on their book. (The sad reality is that this book is much easier than the books I recommended above.)