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An Invitation to Real Analysis Mathematical Association of America 1st Edition by Luis F Moreno ISBN 1939512050 9781939512055

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Author(s): Luis F. Moreno
ISBN(s): 9781939512055, 1939512050
Edition: UK ed.
File Details: PDF, 6.76 MB
Year: 2015
Language: english
SKU: EB-55922880 Category: Tag:

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ISBN 10: 1939512050 
ISBN 13: 9781939512055
Author: Luis F Moreno

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.

An Invitation to Real Analysis Mathematical Association of America 1st Table of contents:

  1. paradoxes?

  2. logical foundations

  3. proof, and the natural numbers

  4. the integers, and the ordered field of rational numbers

  5. induction, and well‑ordering

  6. sets

  7. functions

  8. inverse functions

  9. some subsets of the real numbers

  10. the rational numbers are denumerable

  11. the uncountability of the real numbers

  12. the infinite

  13. the complete, ordered field of real numbers

  14. further properties of real numbers

  15. cluster points and related concepts

  16. the triangle inequality

  17. infinite sequences

  18. limits of sequences

  19. divergence: the non‑existence of a limit

  20. four great theorems in real analysis

  21. limit theorems for sequences

  22. cauchy sequences and the cauchy convergence criterion

  23. the limit superior and limit inferior of a sequence

  24. limits of functions

  25. continuity and discontinuity

  26. the sequential criterion for continuity

  27. theorems about continuous functions

  28. uniform continuity

  29. infinite series of constants

  30. series with positive terms

  31. further tests for series with positive terms

  32. series with negative terms

  33. rearrangements of series

  34. products of series

  35. the numbers e and …

  36. the functions exp x and ln x

  37. the derivative

  38. theorems for derivatives

  39. other derivatives

  40. the mean value theorem

  41. taylor’s theorem

  42. infinite sequences of functions

  43. infinite series of functions

  44. power series

  45. operations with power series

  46. taylor series

  47. taylor series, part ii

  48. the riemann integral

  49. the riemann integral, part ii

  50. the fundamental theorem of integral calculus

  51. improper integrals

  52. the cauchy‑schwarz and minkowski inequalities

  53. metric spaces

  54. functions and limits in metric spaces

  55. some topology of the real number line

  56. the cantor ternary set

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