Overview

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.

Full Product Details

Author:   Luis Moreno
Publisher:   Mathematical Association of America
Imprint:   Mathematical Association of America
Dimensions:   Width: 15.20cm , Height: 4.00cm , Length: 22.90cm
Weight:   1.315kg
ISBN:  

9781939512055

ISBN 10:   1939512050
Pages:   680
Publication Date:   30 December 2015
Audience:   College/higher education ,  Tertiary & Higher Education
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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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. Limits 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–Schwartz 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; 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; Bibliography; Solutions; Index.

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. In his notes to the Instructor, the author writes, With this book I hope to ease a student's transition from what may be called a 'consumer of mathematics' up through calculus II, into one beginning to participate in its creative process. The 55 chapters are quite short and might more properly be called sections, and the author suggests that in a 14-week semester most of the book could be covered. This seems rather ambitious, and it appears to me that it would be more suitable for a two-semester course (in the Canadian system of 12 or 13 weeks each) in second year. Some nice features are a preliminary section on paradoxes, followed by five chapters on foundations, from logic to set theory. After this, all the standard topics of beginning analysis are covered. While quite reasonable the material is largely restricted to analysis on the real line, close to the end there are two chapters on metric spaces, followed by Some Topology of the Real Number Line and The Cantor Ternary Set as the final two chapters. The 30-page appendix contains, among others, a section on Farey sequences and one on continued fractions, topics that are more often found in beginning number theory courses, but are not out of place here. With almost 700 pages, this book is almost as large and heavy as the other three books in this column combined. Still, it is quite reasonably priced, and the value-added features appear to warrant the size. For instance, the approximately 600 exercises give rise to 93 pages of solutions to odd-numbered exercises. The annotated bibliography will be appreciated by both the instructor and by interested students. - CMS Notices

Author Information

Luis F. Moreno received his B.A. in mathematics at Rensselaer Polytechnic Institute in 1973, an M.S. in mathematics education at State University of New York, Albany in 1976, and an M.A. in mathematics at State University of New York, Albany in 1982. He belongs to the Mathematical Association of America (Seaway Section 2nd vice-chair in 2000) and New York State Mathematics Association of Two-Year Colleges, being campus liaison for both organizations. He teaches at Broome Community College where, besides the standard undergraduate courses through linear algebra and real analysis, he has taught courses in statistics, statistical quality control, and logic.

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