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Real Analysis

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Real Analysis

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.

AnalysisMathematics
Author
Eugene Boman & Robert Rogers
Publisher
Pennsylvania State University & SUNY Fredonia
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 45 sections
  1. 01: Numbers - Real () and Rational ()
  2. 1.01: Real and Rational Numbers
  3. 1.E: Numbers - Real (ℝ) and Rational (ℚ) (Exercises)
  4. 02: Calculus in the 17th and 18th Centuries
  5. 2.01: Newton and Leibniz Get Started
  6. 2.02: Power Series as Infinite Polynomials
  7. 2.E: Calculus in the 17th and 18th Centuries (Exercises)
  8. 03: Questions Concerning Power Series
  9. 3.01: Taylor’s Formula
  10. 3.02: Series Anomalies
  11. 3.E: Questions Concerning Power Series (Exercises)
  12. 04: Convergence of Sequences and Series
  13. 4.01: Sequences of Real Numbers
  14. 4.02: The Limit as a Primary Tool
  15. 4.03: Divergence of a Series
  16. 4.E: Convergence of Sequences and Series (Exercises)
  17. 05: Convergence of the Taylor Series- A Tayl of Three Remainders
  18. 5.01: The Integral Form of the Remainder
  19. 5.02: Lagrange’s Form of the Remainder
  20. 5.03: Cauchy’s Form of the Remainder
  21. 5.E: Convergence of the Taylor Series- A “Tayl” of Three Remainders (Exercises)
  22. 06: Continuity - What It Isnt and What It Is
  23. 6.01: An Analytic Definition of Continuity
  24. 6.02: Sequences and Continuity
  25. 6.03: The Definition of the Limit of a Function
  26. 6.04: The Derivative - An Afterthought
  27. 6.E: Continuity - What It Isn’t and What It Is (Exercises)
  28. 07: Intermediate and Extreme Values
  29. 7.01: Completeness of the Real Number System
  30. 7.02: Proof of the Intermediate Value Theorem
  31. 7.03: The Bolzano-Weierstrass Theorem
  32. 7.04: The Supremum and the Extreme Value Theorem
  33. 7.E: Intermediate and Extreme Values (Exercises)
  34. 08: Back to Power Series
  35. 8.01: Uniform Convergence
  36. 8.02: Uniform Convergence- Integrals and Derivatives
  37. 8.03: Radius of Convergence of a Power Series
  38. 8.04: Boundary Issues and Abel’s Theorem
  39. 09: Back to the Real Numbers
  40. 9.01: Trigonometric Series
  41. 9.02: Infinite Sets
  42. 9.03: Cantor’s Theorem and Its Consequences
  43. 10: Epilogue to Real Analysis
  44. 10.01: On the Nature of Numbers
  45. 10.02: Building the Real Numbers