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Analysis: With an Introduction to Proof (4th Edition) Hardcover – December 9, 2004

ISBN-13: 978-0131481015 ISBN-10: 0131481010 Edition: 4th

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Editorial Reviews

Review

"Let me begin by saying that I really like this book, and I do not say that of very many books. What impresses me most is the level of motivation and explanation given for the basic logic, the construction of proofs, and the ways of thinking about proofs that this book provides in its first few sections. It felt that the author was talking to the reader the way I would like to talk to students. There was an air of familiarity there. All kinds of useful remarks were made, the type I would like to make in my lectures." — Aimo Hinkkanen, University of Illinois at Urbana

"The writing style is suitable for our students. It is clear, logical, and concise. The examples are very helpful and well-developed. The topics are thoroughly covered and at the appropriate level for our students. The material is technically accurate, and the pedagogical material is effectively presented." — John Konvalina, University of Nebraska at Omaha

From the Publisher

A solid presentation of the analysis of functions of a real variable -- with special attention on reading and writing proofs. --This text refers to an out of print or unavailable edition of this title.
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Product Details

  • Hardcover: 400 pages
  • Publisher: Pearson; 4 edition (December 9, 2004)
  • Language: English
  • ISBN-10: 0131481010
  • ISBN-13: 978-0131481015
  • Product Dimensions: 8 x 1 x 9.3 inches
  • Shipping Weight: 1.8 pounds
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (20 customer reviews)
  • Amazon Best Sellers Rank: #453,260 in Books (See Top 100 in Books)

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Customer Reviews

The book is very clear in both layout and prose.
Parby Grylam
I'd recommend this book for self study as well as a supplement to any introductory analysis course.
Zachary Turner
I encourage anyone who is using this book to study this chapter ahead of time.
Jetser Carasco

Most Helpful Customer Reviews

60 of 60 people found the following review helpful By Charlie Johnson on September 4, 2002
Format: Hardcover
I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book. For the self-studier, this book is excellent! I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc... For the beginner, this book is superior to all of them. A plethora of examples. Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought. If you already have had some Analysis, then this book is probably not for you. But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat. After this book, one should be able to tackle "Papa Rudin". For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis". This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration. On the whole, this is a great start! If proof-based math is new to you, then you will appreciate the first chapter on proofs. Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration. That's really only nit picking, though.
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52 of 53 people found the following review helpful By Zachary Turner on July 2, 2002
Format: Hardcover
I didn't think this book was going to be very good, but the author has "proved" me wrong ;-) This book starts out so basic that in my class (which was the first analysis course in our math department) we actually skipped the first 1/3 or so of the book. The first 9 or 10 sections consist of stuff like basic set theory, logic, definition of a function, etc. I would think that even the most elementary Analysis books would completely leave this out and expect that the reader is already familiar with this. So if you need it, this book will be a good resource for you.
Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject.
All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis).
The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems.
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25 of 25 people found the following review helpful By Gregory E. Hersh on April 20, 2008
Format: Hardcover Verified Purchase
This is fairly basic introduction to Principles of Analysis, on intermediate undergrad level, strictly in R^1. The only other similar book I'm familiar is Kirkwood. The books of Rudin, Apostol, etc present the subject on much higher level.

My original intention was to take a course with Rudin, but after I've realized I had a hard time digesting his style, I've decided to take more elementary course. I knew the course would be using Lay, so I got this textbook and tried to learn it on my own, but wasn't sure how I was doing and ended up taking the course (still with Lay) anyway. So I'm quite familiar with this textbook. The only topics we didn't cover is "series" and "sequences and series of functions".

Now overall I would say it's a mixed bag. First, the good things. The first few introductory sections on sets and proof techniques are excellent, highly recommended, that's how I learned how to prove. I found exercises very useful.

Now things I don't like. First, lots of typos. I think I had 4th edition, and still I've managed to find over 20 misprints, incorrect references, etc, etc, all were reported directly to author. Second, and that's probably more important, in several instances the proofs are too convoluted and not self-motivating. To be more specific, the proof of Heine-Borell theorem is less than adequate. It is correct, but that's the kind of proof you read and then entirely forget how it went. I remember on the first reading I didn't feel comfortable with this proof at all. When I discussed this book with professor I was going to take that course with, he (surprisingle) agreed with me and told me he would present a different proof (and he did, much better one). Another example: proof that the modified Dirichlet function is Riemann-integrable.
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6 of 7 people found the following review helpful By :) on July 21, 2011
Format: Hardcover Verified Purchase
I love this book. It so easy to understand and will be a helpful guide with my encounters with Anal. this coming semester.
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3 of 3 people found the following review helpful By Parby Grylam on February 6, 2010
Format: Hardcover Verified Purchase
I have the third edition, which I purchased for self study after I ran into trouble in Kolmogorov and Fomin, Introductory Real Analysis, which I had purchased after I ran into trouble with the topology and real analysis assumed by O'Neill in Elementary Differential Geometry. The advantages of Lay's book are described very well in the editorial reviews above. The book is very clear in both layout and prose. The author anticipates questions and explains the reasons for strategems used in proofs. The logical connections among such concepts as open and closed, complete, compact, continuity, metric spaces, and topology are presented clearly. I am enjoying Lay's book and I anticipate that I will soon be resuming study in differential geometry.
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