Invitation to Real Analysis (Mathematical Association of America Textbooks)
By: Luis Moreno (author)Hardback
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Both a stepping stone to higher analysis courses and a foundation for deeper reasoning in applied mathematics, this book provides a broad foundation in real analysis. In connection with this, within the chapters, readers are pointed to numerous accessible articles from The College Mathematics Journal and The American Mathematical Monthly. Axioms are presented with an emphasis on their distinguishing characteristic, culminating with the axioms that define the reals. Set theory is another theme found in this book, running underneath the rigorous development of functions, sequences and series, and ending with chapters on transfinite cardinal numbers and basic point-set topology. Differentiation and integration are developed rigorously with the goal of forming a firm foundation for deeper study. A historical theme interweaves throughout the book, with many quotes and accounts of interest to all readers. Over 600 exercises, dozens of figures, an annotated bibliography, and several appendices help the learning process.
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.
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 gamma; 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.
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