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. Table of Contents 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 ee and ?? 35. The Functions exp xx and ln xx 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 ?nk=0<(1+1n)n+1?k=0n<(1+1n)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
- ISBN: 978-1-93951-205-5
- Editorial: Mathematical Association of America
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