Hardcover ISBN:  9781470449285 
Product Code:  AMSTEXT/36 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
Electronic ISBN:  9781470452339 
Product Code:  AMSTEXT/36.E 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 36; 2019; 303 ppMSC: Primary 26;
This book is an introduction to real analysis for a onesemester 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 proofwriting 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.ReadershipUndergraduate and graduate students interested in learning and teaching undergraduate real analysis.

Table of Contents

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


Additional Material

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This book is an introduction to real analysis for a onesemester 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 proofwriting 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