Introduction to Real Analysis

I have read this whole book for a Phd qualifying exam, mastering all the proofs and solving almost all the excercises, excep for the sections on numerical methods. I can say that this book is a masterpiece. The proofs are clear and easy to follow, and the book flowes smoothly. I can say that it is a classic in its filed as Royden's Real Analysis (3rd Edition), Churchill's Complex Variables and Applications, Fraleigh's First Course in Abstract Algebra, A (7th Edition) and so on.

I'm using this book for my real analysis course at University of Illinois and I love it. Most readers seem to be upset that some of the material isn't presented as easily as it could be, but this book is an introduction to real analysis, not to math. This is not a good book for people who have never written or read formal proofs or who are not familiar with concepts like the triangle inequality. This is a good book if you are familiar with formal mathematics and have interest in real analysis.

Honestly this is a 4 star book, but like many of the advanced math textbooks the average score is too low, because of reviewers who clearly did not understand what they were getting into. Probably the best piece of advice with regards to advanced math books like this is given in the "Preface to the Student" in Sheldon Axler's Linear Algebra Done Right, he states: "You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast." If you study from this book from that standpoint, you will get a lot out of it. But its a serious commitment.

"Introduction to Real Analysis" by R. G. Bartle and D. R. Sherbert gives an excellent introduction to the topic of real analysis. The chapters are presented in a logical order, such that one topic flows seamlessly into the next. The authors explain the concepts of real analysis very clearly and succinctly. This book would be an excellent reference for those currently enroled in a real analysis course, or for those that simply need to brush up on the concepts and ideas of the subject.

A good introduction to real analysis. Proofs are detailed. This book is definitely for anyone who loves real analysis.