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Introduction to Real Analysis [Hardcover]

Robert G. Bartle , Donald R. Sherbert
5.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

Jan. 18 2011 0471433314 978-0471433316 4
This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.

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5.0 out of 5 stars Did the trick Dec 28 2013
Format:Hardcover|Verified Purchase
This book was used as a text book for one of my university classes. I can't say I enjoyed the subject but I did find the book very helpful throughout my course. I really liked the layout, and the examples included.
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Amazon.com: 3.7 out of 5 stars  12 reviews
10 of 10 people found the following review helpful
4.0 out of 5 stars A solid course in real analysis of a single variable Nov. 1 2011
By Vincent Poirier - Published on Amazon.com
Format:Hardcover
Bartle and Sherbert's classic Introduction To Real Analysis gives a rigorous development of real analysis in one variable. Analysis is a branch of mathematics that justifies and proves all the techniques and results of differential & integral calculus. It deals with concepts such as smoothness, convergence, divergence, and so on.

Their treatment of limits, of continuity, of convergence, of differentiation and integration is exact and complete. They give readers a full grounding in epsilon/delta proof methodology for the major theorems of modern single variable calculus.

Because they deal in a single variable, they don't spend much time on basic topology. The book consists of eight chapters. A brief introduction to set theory is followed by a presentation of the real number system. Note that they don't construct the field of real numbers, they merely state the completeness theorem that fills in the gaps found in the field of rational numbers (e.g. the square root of two is a real number not found in the rationals).

The meat of the book begins with chapter three on sequences followed by chapters on limits & continuity, differentiation, Riemann integration, sequences of functions, and finally infinite series.

The many exercises will give readers much opportunity to hone their skills.

I have a few pet peeves. I find the tone a little patronizing. Walter Rudin's Principles of Mathematical Analysis is much more rigorous and explores the topic in greater depth than does Bartle & Sherbert's textbook, but he nowhere adopts their slightly consdescending tone.

Also, the presentation is a little dry. Many of the theorems they give are profound and exciting but one doesn't get this from the text. And they miss out on even hinting at fascinating results because it falls outside the scope of their program. For example, they spend a great deal of time on a rigorous elaboration the sine as cosine functions purely through their derivative properties, with no reference to their geometry interpretation. But because their text doesn't deal with complex numbers, they miss out on presenting a beautiful result that follows straightforwardly from this construction.

Overall, a solid and correct but not very inspiring introduction to the topic. Still, this is a great book from which to teach a course. Teachers can supply the inspiration themselves.

Vincent Poirier, Tokyo
2 of 2 people found the following review helpful
5.0 out of 5 stars Super solid introduction to Real Analysis Nov. 5 2012
By Chan - Published on Amazon.com
Format:Hardcover|Verified Purchase
For undergraduate students, this book is one of the best introduction to Real Analysis. The nice thing about this book is there are many good examples for each Theorem which help you reinforce what you just read. I've been using this book for my first course in Introduction to Analysis, and I'm in love with it. The structure of the book is also very organized, and exercises are very relevant to each chapter. Excellent book for Introduction to Real Analysis.
3 of 4 people found the following review helpful
4.0 out of 5 stars A Good Introduction to Real Analysis Nov. 14 2011
By Nehemiah Leong - Published on Amazon.com
Format:Hardcover
This is, in my opinion, a 4-star book on real analysis at an introductory level. The authors took special care to explain the bolts and nuts of analysis and gradually developed the tools needed to understand the subject. Although the notations can at times be cumbersome, the book is readable on the whole and any beginner in analysis can benefit from it. There is also a brief treatment on Henstock's integration which is obviously one of Bartle's forte. I like the exercises, some of which are accompanied with useful hints and brief solutions. It is thus an excellent book for self-study
5.0 out of 5 stars Solid Introduction to Real Analysis Oct. 24 2013
By Christian Farina - Published on Amazon.com
Format:Hardcover|Verified Purchase
This book provides a solid introduction to real analysis in one variable. The first two chapters introduce the basics of set theory, functions and mathematical induction. Also, the properties of real numbers are introduced here "borrowing" the concept and properties of field from abstract algebra.
The following chapters deal with sequences and series of numbers, limits, continuity, differentiation, integration, sequences and series of function, in this order.
I think the material is presented clearly and the results are proven rigorously throughout the entire book. There are a lot of worked-out examples and many exercises that will test the reader's understanding. Solutions and hints to many (notice, not only the odd ones) of the problems are given in the back of the book. There is also an appendix on logic for those who might need to review the basics, and one on metric spaces and Lebesgue integrals for those students who want to go a bit farther.
In my opinion, this book is not as good as Rudin's book, but it does the job better than many other introductory books on the same topic. For a horrible book see Jiri Lebl's text.
Real analysis is hard, independently of the book you use. It requires a lot of care and hard work. This book does the best it can at clearing the path for you.
4.0 out of 5 stars Hate the material, but the book is okay Oct. 22 2013
By qyu - Published on Amazon.com
Format:Hardcover|Verified Purchase
Hate the material, but the book is okay. Of course there are some typos in the book, but I don't blame it because you can't really avoid those mistakes when writing such complicated book.
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