Introduction to Mathematical Analysis I

by Lafferriere, Lafferriere, and Nguyen

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Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

Description from LibreTexts.

Subjects
Mathematics, Analysis
Licence
CC BY-NC 4.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

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  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. About this Book
  7. 1: Tools for Analysis
  8. 1.1: Basic Concepts of Set Theory
  9. 1.2: Functions
  10. 1.3: The Natural Numbers and Mathematical Induction
  11. 1.4: Ordered Field Axioms
  12. 1.5: The Completeness Axiom for the Real Numbers
  13. 1.6: Applications of the Completeness Axiom
  14. 2: Sequences
  15. 2.1: Convergence
  16. 2.2: Limit Theorems
  17. 2.3: Monotone Sequences
  18. 2.4: The Bolazno-Weierstrass Theorem
  19. 2.5: Limit Superior and Limit Inferior
  20. 2.6: Open Sets, Closed Sets, Compact Sets, and Limit Points
  21. 3: Limits and Continuity
  22. 3.1: Limits of Functions
  23. 3.2: Limit Theorems
  24. 3.3: Continuity
  25. 3.4: Properties of Continuous Functions
  26. 3.5: Uniform Continuity
  27. 3.6: Limit Superior and Limit Inferior of Functions
  28. 3.7: Lower Semicontinuity and Upper Semicontinuity
  29. 4: Differentiation
  30. 4.1: Definition and Basic Properties of the Derivative
  31. 4.2: THE MEAN VALUE THEOREM
  32. 4.3: SOME APPLICATIONS OF THE MEAN VALUE THEOREM
  33. 4.4: L'Hopital's Rule
  34. 4.5: Taylor's Theorem
  35. 4.6: CONVEX FUNCTIONS AND DERIVATIVES
  36. 4.7: NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS
  37. 5: Solutions and Hints for Selected Exercises
  38. 5.1: CHAPTER 1
  39. 5.2: CHAPTER 2
  40. 5.3: CHAPTER 3
  41. 5.4: CHAPTER 4
  42. Back Matter
  43. Index
  44. Glossary
  45. Detailed Licensing