Invitation to Real Analysis

Luis Moreno (Author)

9781939512055

Hardback, published 30 December 2015

680 pages
22.9 x 15.2 x 4 cm, 1.315 kg

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… The annotated bibliography will be appreciated by both the instructor and by interested students." - CMS Notices

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.

• 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-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
• Appendices A-F
• Annotated Bibliography
• Solutions to Odd-Numbered Exercises
• Index