Invitation to Real Analysis

A broad and rigorous treatment of real analysis and a platform to higher mathematics, with historical material and 600 exercises.

Luis F. Moreno (Author)

9781939512055, The Mathematical Association of America

Hardback, published 9 July 2015

678 pages
26.1 x 18.2 x 4 cm, 1.32 kg

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.

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.

Subject Areas: Real analysis, real variables [PBKB], Calculus & mathematical analysis [PBK], Set theory [PBCH], Mathematics [PB]