*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.

*Solutions manuals available upon request. Please contact: Carol Baxter at cbaxter@maa.org.*

To the Student

To the Instructor

0. Paradoxes?

1. Logical Foundations

2. Proof, and the Natural Numbers

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

4. Induction and Well-Ordering

5. Sets

6. Functions

7. Inverse Functions

8. Some Subsets of the Real Numbers

9. The Rational Numbers are Denumerable

10. The Uncountability of the Real Numbers

11. The Infinite

12. The Complete, Ordered Field of Real Numbers

13. Further Properties of Real Numbers

14. Cluster Points and Related Concepts

15. The Triangle Inequality

16. Infinite Sequences

17. Limit of Sequences

18. Divergence: The Non-Existence of a Limit

19. Four Great Theorems in Real Analysis

20. Limit Theorems for Sequences

21. Cauchy Sequences and the Cauchy Convergence Criterion

22. The Limit Superior and Limit Inferior of a Sequence

23. Limits of Functions

24. Continuity and Discontinuity

25. The Sequential Criterion for Continuity

26. Theorems about Continuous Functions

27. Uniform Continuity

28. Infinite Series of Constants

29. Series with Positive Terms

30. Further Tests for Series with Positive Terms

31. Series with Negative Terms

32. Rearrangements of Series

33. Products of Series

34. The Numbers \(e\) and \(γ\)

35. The Functions exp \(x\) and ln \(x\)

36. The Derivative

37. Theorems for Derivatives

38. Other Derivatives

39. The Mean Value Theorem

40. Taylor’s Theorem

41. Infinite Sequences of Functions

42. Infinite Series of Functions

43. Power Series

44. Operations with Power Series

45. Taylor Series

46. Taylor Series, Part II

47. The Riemann Integral

48. The Riemann Integral, Part II

49. The Fundamental Theorem of Integral Calculus

50. Improper Integrals

51. The Cauchy-Schwarz and Minkowski Inequalities

52. Metric Spaces

53. Functions and Limits in Metric Spaces

54. Some Topology of the Real Number Line

55. The Cantor Ternary Set

Appendix A: Farey Sequences

Appendix B: Proving that \(\sum_{k=0}^{n} < (1 + \frac{1}{n})^{n+1}\)

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

Annotated Bibliography

Solutions to Odd-Numbered Exercises

Index