Invitation to Real Analysis

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 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.