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Analysis: With an Introduction to Proof (4th Edition) [Hardcover]

Steven R. Lay
4.1 out of 5 stars  See all reviews (19 customer reviews)

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Analysis with an Introduction to Proof (5th Edition) (Featured Titles for Real Analysis) Analysis with an Introduction to Proof (5th Edition) (Featured Titles for Real Analysis) 3.7 out of 5 stars (6)
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Book Description

December 9, 2004 0131481010 978-0131481015 4

By introducing logic and by emphasizing the structure and nature of the arguments used, this book helps readers transition from computationally oriented mathematics to abstract mathematics with its emphasis on proofs. Uses clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers. Offers a new boxed review of key terms after each section. Rewrites many exercises. Features more than 250 true/false questions. Includes more than 100 practice problems. Provides exceptionally high-quality drawings to illustrate key ideas. Provides numerous examples and more than 1,000 exercises. A thorough reference for readers who need to increase or brush up on their advanced mathematics skills.

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Editorial Reviews


"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: 0.8 x 8.1 x 9.2 inches
  • Shipping Weight: 1.8 pounds
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (19 customer reviews)
  • Amazon Best Sellers Rank: #86,659 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
60 of 60 people found the following review helpful
4.0 out of 5 stars Definitely a good first text September 4, 2002
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
4.0 out of 5 stars This book was surprisingly good July 2, 2002
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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24 of 24 people found the following review helpful
3.0 out of 5 stars Acceptable but could have been better. 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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8 of 9 people found the following review helpful
5.0 out of 5 stars Great Book for Intro to Analysis March 13, 2008
This is a very good book for someone to look at before going into an analysis class with Rudin. If you have never done proofs or seen metric spaces or uniform continuity, etc., this is a nice, but brief, intro. This book will NOT teach you analysis - you have to use Rudin for that. But it is great for acquainting/preparing you for Rudin.
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Most Recent Customer Reviews
3.0 out of 5 stars Meh
Interaction with the buyer was great.
The book itself is not something I would choose other than for my class. Not my style of learning.
Published 6 months ago by Ellie Woods
5.0 out of 5 stars was in very good condition
Love this book since it explains the proofs of the theorems well. Great book to learn analysis for a beginner.
Published 9 months ago by Nirosha
1.0 out of 5 stars Useless
Not enough examples of actual proofs and meager explanations for answers to examples. This book is unhelpful in nearly every way. Read more
Published 12 months ago by James Wymer
4.0 out of 5 stars Good quality
Obviously a used book, because there are marks and scratches on pages. Generally like a new book, which is pretty much like what it is described.
Published 17 months ago by Sean
5.0 out of 5 stars Get this book
if you taking Analysis get this one, because you are going to need it for the class in advanced math.
Published 18 months ago by David
5.0 out of 5 stars Good!
This book is very good for new students of proofing things. It is good for us to choose this book for analysis.
Published 21 months ago by Lawrence
5.0 out of 5 stars Real Anal.
I love this book. It so easy to understand and will be a helpful guide with my encounters with Anal. this coming semester.
Published on July 21, 2011 by :)
4.0 out of 5 stars great intro to analysis
Mathematical analysis is difficult to understand and this book explains everything in great detail. Some of the practice questions are pretty challenging, but not impossible. Read more
Published on April 19, 2011 by Travis W. Richardson
2.0 out of 5 stars ugh...
Steven Lay is every bad math teacher you've ever had rolled into one giant ball of smarm and uselessness. Read more
Published on October 7, 2010 by Peter Johnson
4.0 out of 5 stars Thumbs up!
Money saver! got this book for about 1/4 of the asking price at my local university.
Published on September 27, 2010 by 0xcavalieri
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