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An Invitation to Real Analysis

An Invitation to Real Analysis

Luis F. Moreno

Catalog Code: IRA
Print ISBN: 978-1-93951-205-5
Electronic ISBN: 978-1-61444-617-0
680 pp., Hardbound, 2015
List Price: $75.00
Member Price: $56.25
Series: MAA Textbooks

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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 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. Limit 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-Schwarz 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 \(\sum_{k=0}^{n} < (1 + \frac{1}{n})^{n+1}\)
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
Annotated Bibliography
Solutions to Odd-Numbered Exercises
Index

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