Hardcover ISBN:  9781939512055 
Product Code:  TEXT/31 
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AMS Member Price:  $56.25 
eBook ISBN:  9781614446170 
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AMS Member Price:  $51.75 
Hardcover ISBN:  9781939512055 
eBook: ISBN:  9781614446170 
Product Code:  TEXT/31.B 
List Price:  $144.00 $109.50 
MAA Member Price:  $108.00 $82.13 
AMS Member Price:  $108.00 $82.13 
Hardcover ISBN:  9781939512055 
Product Code:  TEXT/31 
List Price:  $75.00 
MAA Member Price:  $56.25 
AMS Member Price:  $56.25 
eBook ISBN:  9781614446170 
Product Code:  TEXT/31.E 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
Hardcover ISBN:  9781939512055 
eBook ISBN:  9781614446170 
Product Code:  TEXT/31.B 
List Price:  $144.00 $109.50 
MAA Member Price:  $108.00 $82.13 
AMS Member Price:  $108.00 $82.13 

Book DetailsAMS/MAA TextbooksVolume: 31; 2015; 661 pp
An Invitation to Real Analysis is written both as a stepping stone to higher calculus and analysis courses, and as foundation for deeper reasoning in applied mathematics. This book also provides a broader foundation in real analysis than is typical for future teachers of secondary mathematics. In connection with this, within the chapters, students are pointed to numerous articles from The College Mathematics Journal and The American Mathematical Monthly. These articles are inviting in their level of exposition and their wideranging content. Axioms are presented with an emphasis on the distinguishing characteristics that new ones bring, culminating with the axioms that define the reals.
Set theory is another theme found in this book, beginning with what students are familiar with from basic calculus. This theme runs underneath the rigorous development of functions, sequences, and series, and then ends with a chapter on transfinite cardinal numbers and with chapters on basic pointset topology. Differentiation and integration are developed with the standard level of rigor, but always with the goal of forming a firm foundation for the student who desires to pursue deeper study.
A historical theme interweaves throughout the book, with many quotes and accounts of interest to all readers. Over 600 exercises and dozens of figures help the learning process. Several topics (continued fractions, for example), are included in the appendices as enrichment material. An annotated bibliography is included.
Ancillaries:

Table of Contents

Chapters

Chapter 0. Paradoxes?

Chapter 1. Logical Foundations

Chapter 2. Proof, and the Natural Numbers

Chapter 3. The Integers, and the Ordered Field of Rational Numbers

Chapter 4. Induction and WellOrdering

Chapter 5. Sets

Chapter 6. Functions

Chapter 7. Inverse Functions

Chapter 8. Some Subsets of the Real Numbers

Chapter 9. The Rational Numbers Are Denumerable

Chapter 10. The Uncountability of the Real Numbers

Chapter 11. The Infinite

Chapter 12. The Complete, Ordered Field of Real Numbers

Chapter 13. Further Properties of Real Numbers

Chapter 14. Cluster Points and Related Concepts

Chapter 15. The Triangle Inequality

Chapter 16. Infinite Sequences

Chapter 17. Limits of Sequences

Chapter 18. Divergence: The NonExistence of a Limit

Chapter 19. Four Great Theorems in Real Analysis

Chapter 20. Limit Theorems for Sequences

Chapter 21. Cauchy Sequences and the Cauchy Convergence Criterion

Chapter 22. The Limit Superior and Limit Inferior of a Sequence

Chapter 23. Limits of Functions

Chapter 24. Continuity and Discontinuity

Chapter 25. The Sequential Criterion for Continuity

Chapter 26. Theorems About Continuous Functions

Chapter 27. Uniform Continuity

Chapter 28. Infinite Series of Constants

Chapter 29. Series with Positive Terms

Chapter 30. Further Tests for Series with Positive Terms

Chapter 31. Series with Negative Terms

Chapter 32. Rearrangements of Series

Chapter 33. Products of Series

Chapter 34. The Numbers $e$ and $y$

Chapter 35. The Functions exp $x$ and ln $x$

Chapter 36. The Derivative

Chapter 37. Theorems for Derivatives

Chapter 38. Other Derivatives

Chapter 39. The Mean Value Theorem

Chapter 40. Taylor’s Theorem

Chapter 41. Infinite Sequences of Functions

Chapter 42. Infinite Series of Functions

Chapter 43. Power Series

Chapter 44. Operations with Power Series

Chapter 45. Taylor Series

Chapter 46. Taylor Series, Part II

Chapter 47. The Riemann Integral

Chapter 48. The Riemann Integral, Part II

Chapter 49. The Fundamental Theorem of Integral Calculus

Chapter 50. Improper Integrals

Chapter 51. The CauchySchwarz and Minkowski Inequalities

Chapter 52. Metric Spaces

Chapter 53. Functions and Limits in Metric Spaces

Chapter 54. Some Topology of the Real Number Line

Chapter 55. The Cantor Ternary Set

Appendix A. Farey Sequences

Appendix B. Proving that $\sum ^n_{k=0}\frac {1}{k!}<(1+\frac {1}{n})^{n+1}$

Appendix C. The Ruler Function Is Riemann Integrable

Appendix D. Continued Fractions

Appendix E. L’Hospital’s Rule

Appendix F. Symbols, and the Greek Alphabet


Additional Material

Reviews

The title of this book suggests a friendly tone and a gentle introduction to real analysis. This does indeed seem to be the case, as the book's size and readerfriendly layout suggest. ... The annotated bibliography will be appreciated by both the instructor and by interested students.
CMS Notices


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 Book Details
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An Invitation to Real Analysis is written both as a stepping stone to higher calculus and analysis courses, and as foundation for deeper reasoning in applied mathematics. This book also provides a broader foundation in real analysis than is typical for future teachers of secondary mathematics. In connection with this, within the chapters, students are pointed to numerous articles from The College Mathematics Journal and The American Mathematical Monthly. These articles are inviting in their level of exposition and their wideranging content. Axioms are presented with an emphasis on the distinguishing characteristics that new ones bring, culminating with the axioms that define the reals.
Set theory is another theme found in this book, beginning with what students are familiar with from basic calculus. This theme runs underneath the rigorous development of functions, sequences, and series, and then ends with a chapter on transfinite cardinal numbers and with chapters on basic pointset topology. Differentiation and integration are developed with the standard level of rigor, but always with the goal of forming a firm foundation for the student who desires to pursue deeper study.
A historical theme interweaves throughout the book, with many quotes and accounts of interest to all readers. Over 600 exercises and dozens of figures help the learning process. Several topics (continued fractions, for example), are included in the appendices as enrichment material. An annotated bibliography is included.
Ancillaries:

Chapters

Chapter 0. Paradoxes?

Chapter 1. Logical Foundations

Chapter 2. Proof, and the Natural Numbers

Chapter 3. The Integers, and the Ordered Field of Rational Numbers

Chapter 4. Induction and WellOrdering

Chapter 5. Sets

Chapter 6. Functions

Chapter 7. Inverse Functions

Chapter 8. Some Subsets of the Real Numbers

Chapter 9. The Rational Numbers Are Denumerable

Chapter 10. The Uncountability of the Real Numbers

Chapter 11. The Infinite

Chapter 12. The Complete, Ordered Field of Real Numbers

Chapter 13. Further Properties of Real Numbers

Chapter 14. Cluster Points and Related Concepts

Chapter 15. The Triangle Inequality

Chapter 16. Infinite Sequences

Chapter 17. Limits of Sequences

Chapter 18. Divergence: The NonExistence of a Limit

Chapter 19. Four Great Theorems in Real Analysis

Chapter 20. Limit Theorems for Sequences

Chapter 21. Cauchy Sequences and the Cauchy Convergence Criterion

Chapter 22. The Limit Superior and Limit Inferior of a Sequence

Chapter 23. Limits of Functions

Chapter 24. Continuity and Discontinuity

Chapter 25. The Sequential Criterion for Continuity

Chapter 26. Theorems About Continuous Functions

Chapter 27. Uniform Continuity

Chapter 28. Infinite Series of Constants

Chapter 29. Series with Positive Terms

Chapter 30. Further Tests for Series with Positive Terms

Chapter 31. Series with Negative Terms

Chapter 32. Rearrangements of Series

Chapter 33. Products of Series

Chapter 34. The Numbers $e$ and $y$

Chapter 35. The Functions exp $x$ and ln $x$

Chapter 36. The Derivative

Chapter 37. Theorems for Derivatives

Chapter 38. Other Derivatives

Chapter 39. The Mean Value Theorem

Chapter 40. Taylor’s Theorem

Chapter 41. Infinite Sequences of Functions

Chapter 42. Infinite Series of Functions

Chapter 43. Power Series

Chapter 44. Operations with Power Series

Chapter 45. Taylor Series

Chapter 46. Taylor Series, Part II

Chapter 47. The Riemann Integral

Chapter 48. The Riemann Integral, Part II

Chapter 49. The Fundamental Theorem of Integral Calculus

Chapter 50. Improper Integrals

Chapter 51. The CauchySchwarz and Minkowski Inequalities

Chapter 52. Metric Spaces

Chapter 53. Functions and Limits in Metric Spaces

Chapter 54. Some Topology of the Real Number Line

Chapter 55. The Cantor Ternary Set

Appendix A. Farey Sequences

Appendix B. Proving that $\sum ^n_{k=0}\frac {1}{k!}<(1+\frac {1}{n})^{n+1}$

Appendix C. The Ruler Function Is Riemann Integrable

Appendix D. Continued Fractions

Appendix E. L’Hospital’s Rule

Appendix F. Symbols, and the Greek Alphabet

The title of this book suggests a friendly tone and a gentle introduction to real analysis. This does indeed seem to be the case, as the book's size and readerfriendly layout suggest. ... The annotated bibliography will be appreciated by both the instructor and by interested students.
CMS Notices