Bibliografía

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 ContentsTo the StudentTo the Instructor0. Paradoxes?1. Logical Foundations2. Proof, and the Natural Numbers3. The Integers, and the Ordered Field of Rational Numbers4. Induction and Well-Ordering5. Sets6. Functions7. Inverse Functions8. Some Subsets of the Real Numbers9. The Rational Numbers are Denumerable10. The Uncountability of the Real Numbers11. The Infinite12. The Complete, Ordered Field of Real Numbers13. Further Properties of Real Numbers14. Cluster Points and Related Concepts15. The Triangle Inequality16. Infinite Sequences17. Limit of Sequences18. Divergence: The Non-Existence of a Limit19. Four Great Theorems in Real Analysis20. Limit Theorems for Sequences21. Cauchy Sequences and the Cauchy Convergence Criterion22. The Limit Superior and Limit Inferior of a Sequence23. Limits of Functions24. Continuity and Discontinuity25. The Sequential Criterion for Continuity26. Theorems about Continuous Functions27. Uniform Continuity28. Infinite Series of Constants29. Series with Positive Terms30. Further Tests for Series with Positive Terms31. Series with Negative Terms32. Rearrangements of Series33. Products of Series34. The Numbers ee and ??35. The Functions exp xx and ln xx36. The Derivative37. Theorems for Derivatives38. Other Derivatives39. The Mean Value Theorem40. Taylor’s Theorem41. Infinite Sequences of Functions42. Infinite Series of Functions43. Po