Real Analysis

by Eugene Boman & Robert Rogers

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The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

Description from LibreTexts.

Publisher
Pennsylvania State University & SUNY Fredonia
Subjects
Mathematics, Analysis
Licence
CC BY-NC-SA 4.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. Prelude to Real Analysis
  3. TitlePage
  4. InfoPage
  5. TitlePage
  6. InfoPage
  7. Table of Contents
  8. Licensing
  9. Table of Contents
  10. 1: Numbers - Real (ℝ) and Rational (ℚ)
  11. 1.1: Real and Rational Numbers
  12. 1.E: Numbers - Real (ℝ) and Rational (ℚ) (Exercises)
  13. 2: Calculus in the 17th and 18th Centuries
  14. 2.1: Newton and Leibniz Get Started
  15. 2.2: Power Series as Infinite Polynomials
  16. 2.E: Calculus in the 17th and 18th Centuries (Exercises)
  17. 3: Questions Concerning Power Series
  18. 3.1: Taylor’s Formula
  19. 3.2: Series Anomalies
  20. 3.E: Questions Concerning Power Series (Exercises)
  21. 4: Convergence of Sequences and Series
  22. 4.1: Sequences of Real Numbers
  23. 4.2: The Limit as a Primary Tool
  24. 4.3: Divergence of a Series
  25. 4.E: Convergence of Sequences and Series (Exercises)
  26. 5: Convergence of the Taylor Series- A “Tayl” of Three Remainders
  27. 5.1: The Integral Form of the Remainder
  28. 5.2: Lagrange’s Form of the Remainder
  29. 5.3: Cauchy’s Form of the Remainder
  30. 5.E: Convergence of the Taylor Series- A “Tayl” of Three Remainders (Exercises)
  31. 6: Continuity - What It Isn’t and What It Is
  32. 6.1: An Analytic Definition of Continuity
  33. 6.2: Sequences and Continuity
  34. 6.3: The Definition of the Limit of a Function
  35. 6.4: The Derivative - An Afterthought
  36. 6.E: Continuity - What It Isn’t and What It Is (Exercises)
  37. 7: Intermediate and Extreme Values
  38. 7.1: Completeness of the Real Number System
  39. 7.2: Proof of the Intermediate Value Theorem
  40. 7.3: The Bolzano-Weierstrass Theorem
  41. 7.4: The Supremum and the Extreme Value Theorem
  42. 7.E: Intermediate and Extreme Values (Exercises)
  43. 8: Back to Power Series
  44. 8.1: Uniform Convergence
  45. 8.2: Uniform Convergence- Integrals and Derivatives
  46. 8.3: Radius of Convergence of a Power Series
  47. 8.4: Boundary Issues and Abel’s Theorem
  48. 9: Back to the Real Numbers
  49. 9.1: Trigonometric Series
  50. 9.2: Infinite Sets
  51. 9.3: Cantor’s Theorem and Its Consequences
  52. 10: Epilogue to Real Analysis
  53. 10.1: On the Nature of Numbers
  54. 10.2: Building the Real Numbers
  55. Back Matter
  56. Index
  57. Index
  58. Glossary
  59. Detailed Licensing