Introduction to Analysis (Pure and Applied Undergraduate Texts) by Edward D. Gaughan (PDF)

Description

Introduction to Analysis is designed to bridge the gap between the intuitive calculus usually offered at the undergraduate level and the sophisticated analysis courses the student encounters at the graduate level. In this book the student is given the vocabulary and facts necessary for further study in analysis. The course for which it is designed is usually offered at the junior level, and it is assumed that the student has little or no previous experience with proofs in analysis. A considerable amount of time is spent motivating the theorems and proofs and developing the reader’s intuition. Of course, that intuition must be tempered with the realization that rigorous proofs are required for theorems. The topics are quite standard: convergence of sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, power series, and convergence of sequences of functions. Many examples are given to illustrate the theory, and exercises at the end of each chapter are keyed to each section. Also, at the end of each section, one finds several Projects. The purpose of a Project is to give the reader a substantial mathematical problem and the necessary guidance to solve that problem. A Project is distinguished from an exercise in that the solution of a Project is a multi-step process requiring assistance for the beginner student.

User’s Reviews

Editorial Reviews: About the Author Ph.D., University of Kansas

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book broke me into analysis. It gives the reader a gentle introduction to the subject. The proofs are rigorous and easy to follow. Before embarking upon an analysis class, a person should take an introduction to proof class. Without such a class, the language of analysis will make no sense to the reader. For example, if you do not know how to do proof by induction, then you will struggle in analysis. The negative reviews for this book likely stem from readers who did not take such a class. Thanks to this book and Munkres’ topology, I am now able to read and enjoy Baby Rudin. Baby Rudin is good, but I would not have liked it without first using Munkres and Gaughan. Note that Gaughan is not a complete substitute for Rudin. Rudin is general whereas Gaughan focuses mainly on the real line. Albeit, Gaughan is much easier on a reader who is new to the subject. Gaughan’s treatment of sequences sets the stage for the book. He then goes into continuity and uniform continuity. The definitions and theorems are lucid. Gaughan then delves into differentiation, integration, and power series. The ordering of the topics allows the reader to move from topic to topic without having to flip back and forth through the book.

⭐Though I don’t like bringing in the atmosphere in which the book is used, intro to analysis really suggests that this is going to be a students first exposure to rigorously enforced proofs. The book, therefore, is probably going to be hard to understand, and with a book that’s so concise about it’s explanations really can only be used in a setting where the teacher is going to provide the foundations of understanding the symbols and logic behind the proofs the book tends to breeze through. The class I was in took so long to understand the concepts we hardly made it anywhere in this book. So in a class where you’re lectures expose you to the terminology and general shape of the proofs so that you can decipher the book I’m sure this could be used. If you’re not getting that in your class maybe you’ll want to find some additional reference material that will walk through it a bit slower.Mindless complaint: Really wish authors wouldn’t start with a Chapter 0. It makes it feel so unimportant.

⭐ES UNA MUY BUENA INTRODUCCIÓN AL ANALYSIS REAL. Es un poco antiguo, pero cumple su objetivo. Existe una edición posterior, con otra editorial que no amplia el tema demasiado.I greatly appreciate the fluency of the logic and discussion found in this book. It was fantastic. I’ve attempted this course at another school using the professors notes (it was death). This book is so much easy to use and understand; it was a night and day difference. There seem to be a lot of different methods for proving these basic theorems inherent to the concepts and principles underlying calculus and this book is very clear and smart about accomplishing this task. I would recommend this book and I am not going to sell it. It is great.

⭐School book

⭐Fast delivery and good.

⭐I had to get this book for my advanced calculus class. There are 3 issues I have with this book. First, that the material in any given chapter gives what is needed to work most, BUT NOT ALL, of the problems at the end of the given chapter. Second, having solutions to selected exercises is a good tool to have, and if an instructor disagrees, he or she can just assign problems that don’t have solutions given. Not all students would use those solutions, but it’s nice to have to verify one is on the right track. Third, the author does not explain things as clearly as other authors do. He (assuming Edward is male) could have explained things much better without adding a single word to the text. I find myself frequently referring to ‘How To Prove It’ by Velleman, since Velleman explains things much better. In short, this book is not the worst I’ve seen, and I would give this book 3 stars, were it not for the price.

⭐Good

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