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An Invitation to Real Analysis
 
An Invitation to Real Analysis
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-93951-205-5
Product Code:  TEXT/31
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-61444-617-0
Product Code:  TEXT/31.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Hardcover ISBN:  978-1-93951-205-5
eBook: ISBN:  978-1-61444-617-0
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
An Invitation to Real Analysis
Click above image for expanded view
An Invitation to Real Analysis
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-93951-205-5
Product Code:  TEXT/31
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-61444-617-0
Product Code:  TEXT/31.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Hardcover ISBN:  978-1-93951-205-5
eBook ISBN:  978-1-61444-617-0
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 Details
     
     
    AMS/MAA Textbooks
    Volume: 312015; 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 wide-ranging 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 point-set 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 Well-Ordering
    • 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 Non-Existence 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 Cauchy-Schwarz 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 reader-friendly layout suggest. ... The annotated bibliography will be appreciated by both the instructor and by interested students.

      CMS Notices
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 312015; 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 wide-ranging 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 point-set 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 Well-Ordering
  • 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 Non-Existence 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 Cauchy-Schwarz 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 reader-friendly layout suggest. ... The annotated bibliography will be appreciated by both the instructor and by interested students.

    CMS Notices
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.