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Invitation to Real Analysis

Cesar E. Silva Williams College, Williamstown, MA
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Hardcover ISBN: 978-1-4704-4928-5
Product Code: AMSTEXT/36
List Price: $99.00 MAA Member Price:$89.10
AMS Member Price: $79.20 Electronic ISBN: 978-1-4704-5233-9 Product Code: AMSTEXT/36.E List Price:$99.00
MAA Member Price: $89.10 AMS Member Price:$79.20
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List Price: $148.50 MAA Member Price:$133.65
AMS Member Price: $118.80 Click above image for expanded view Invitation to Real Analysis Cesar E. Silva Williams College, Williamstown, MA Available Formats:  Hardcover ISBN: 978-1-4704-4928-5 Product Code: AMSTEXT/36  List Price:$99.00 MAA Member Price: $89.10 AMS Member Price:$79.20
 Electronic ISBN: 978-1-4704-5233-9 Product Code: AMSTEXT/36.E
 List Price: $99.00 MAA Member Price:$89.10 AMS Member Price: $79.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$148.50 MAA Member Price: $133.65 AMS Member Price:$118.80
• Book Details

Pure and Applied Undergraduate Texts
Volume: 362019; 303 pp
MSC: Primary 26;

This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions.

A theme in the book is to give more than one proof for interesting facts; this illustrates how different ideas interact and it makes connections among the facts that are being learned. Metric spaces are introduced early in the book, but there are instructions on how to avoid metric spaces for the instructor who wishes to do so. There are questions that check the readers' understanding of the material, with solutions provided at the end. Topics that could be optional or assigned for independent reading include the Cantor function, nowhere differentiable functions, the Gamma function, and the Weierstrass theorem on approximation by continuous functions.

Undergraduate and graduate students interested in learning and teaching undergraduate real analysis.

• Cover
• Title page
• Preface
• Chapter 0. Preliminaries: Sets, Functions, and Induction
• 0.1. Notation on Sets and Functions
• 0.2. Basic Logic: Statements and Logical Connectives
• 0.3. Sets
• 0.4. Functions
• 0.5. Mathematical Induction
• 0.6. More on Sets: Axioms and Constructions
• Chapter 1. The Real Numbers and the Completeness Property
• 1.1. Field and Order Properties of \R
• 1.2. Completeness Property of \R
• 1.3. Countable and Uncountable Sets
• 1.4. Construction of the Real Numbers
• 1.5. The Complex Numbers
• Chapter 2. Sequences
• 2.1. Limits of Sequences
• 2.2. Three Consequences of Order Completeness
• 2.3. The Cauchy Property for Sequences
• Chapter 3. Topology of the Real Numbers and Metric Spaces
• 3.1. Metrics
• 3.2. Open and Closed Sets in \R
• 3.3. Open and Closed Sets in Metric Spaces
• 3.4. Compactness in \R
• 3.5. The Cantor Set
• 3.6. Connected Sets in \R
• 3.7. Compactness, Connectedness, and Completeness in Metric Spaces
• Chapter 4. Continuous Functions
• 4.1. Continuous Functions on \R
• 4.2. Intermediate Value and Extreme Value Theorems
• 4.3. Limits
• 4.4. Uniform Continuity
• 4.5. Continuous Functions on Metric Spaces
• Chapter 5. Differentiable Functions
• 5.1. Differentiable Functions on \R
• 5.2. Mean Value Theorem
• 5.3. Taylor’s Theorem
• Chapter 6. Integration
• 6.1. The Riemann Integral
• 6.2. The Fundamental Theorem of Calculus
• 6.3. Improper Riemann Integrals
• Chapter 7. Series
• 7.1. Series of Real Numbers
• 7.2. Alternating Series and Absolute Convergence
• Chapter 8. Sequences and Series of Functions
• 8.1. Pointwise Convergence
• 8.2. Uniform Convergence
• 8.3. Series of Functions
• 8.4. Power Series
• 8.5. Taylor Series
• 8.6. Weierstrass Approximation Theorem
• 8.7. The Complex Exponential
• Appendix A. Solutions to Questions
• Bibliographical Notes
• Bibliography
• Index
• Back Cover

• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 362019; 303 pp
MSC: Primary 26;

This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions.

A theme in the book is to give more than one proof for interesting facts; this illustrates how different ideas interact and it makes connections among the facts that are being learned. Metric spaces are introduced early in the book, but there are instructions on how to avoid metric spaces for the instructor who wishes to do so. There are questions that check the readers' understanding of the material, with solutions provided at the end. Topics that could be optional or assigned for independent reading include the Cantor function, nowhere differentiable functions, the Gamma function, and the Weierstrass theorem on approximation by continuous functions.

Undergraduate and graduate students interested in learning and teaching undergraduate real analysis.

• Cover
• Title page
• Preface
• Chapter 0. Preliminaries: Sets, Functions, and Induction
• 0.1. Notation on Sets and Functions
• 0.2. Basic Logic: Statements and Logical Connectives
• 0.3. Sets
• 0.4. Functions
• 0.5. Mathematical Induction
• 0.6. More on Sets: Axioms and Constructions
• Chapter 1. The Real Numbers and the Completeness Property
• 1.1. Field and Order Properties of \R
• 1.2. Completeness Property of \R
• 1.3. Countable and Uncountable Sets
• 1.4. Construction of the Real Numbers
• 1.5. The Complex Numbers
• Chapter 2. Sequences
• 2.1. Limits of Sequences
• 2.2. Three Consequences of Order Completeness
• 2.3. The Cauchy Property for Sequences
• Chapter 3. Topology of the Real Numbers and Metric Spaces
• 3.1. Metrics
• 3.2. Open and Closed Sets in \R
• 3.3. Open and Closed Sets in Metric Spaces
• 3.4. Compactness in \R
• 3.5. The Cantor Set
• 3.6. Connected Sets in \R
• 3.7. Compactness, Connectedness, and Completeness in Metric Spaces
• Chapter 4. Continuous Functions
• 4.1. Continuous Functions on \R
• 4.2. Intermediate Value and Extreme Value Theorems
• 4.3. Limits
• 4.4. Uniform Continuity
• 4.5. Continuous Functions on Metric Spaces
• Chapter 5. Differentiable Functions
• 5.1. Differentiable Functions on \R
• 5.2. Mean Value Theorem
• 5.3. Taylor’s Theorem
• Chapter 6. Integration
• 6.1. The Riemann Integral
• 6.2. The Fundamental Theorem of Calculus
• 6.3. Improper Riemann Integrals
• Chapter 7. Series
• 7.1. Series of Real Numbers
• 7.2. Alternating Series and Absolute Convergence
• Chapter 8. Sequences and Series of Functions
• 8.1. Pointwise Convergence
• 8.2. Uniform Convergence
• 8.3. Series of Functions
• 8.4. Power Series
• 8.5. Taylor Series
• 8.6. Weierstrass Approximation Theorem
• 8.7. The Complex Exponential
• Appendix A. Solutions to Questions
• Bibliographical Notes
• Bibliography
• Index
• Back Cover
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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