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The Book of Real Analysis

(Currently in Progress)

1 Characterizing ℝ

  1. ℝ is a Field
  2. ℝ is an Ordered Field
  3. ℝ is a Complete Ordered Field
  4. Working with Suprema and Infima
  5. Existence of Roots
  6. The Archimedean Principle
  7. The Triangle Inequality
  8. The Cauchy-Schwarz Inequality
  9. The Minkowski Inequality
  10. The Extended Real Number System

2 Cardinality and Infinity

Content Under Construction

3 Sequences

  1. The Limit Superior and Limit Inferior of a Sequence
  2. Properties of the Limit Superior and Limit Inferior

4 Series

Content Under Construction

5 The Topology of ℝ

  1. Why Topology Matters
  2. Metric Spaces
  3. Open Sets
  4. Closed Sets
  5. Unions and Intersections of Sets
  6. Further Topological Properties of Real Numbers
  7. Closure of a Set
  8. Connected Sets

6 Limits and Continuity

Content Under Construction

7 Differentiation

Content Under Construction

8 Integration

Content Under Construction

9 Sequences and Series of Functions

Content Under Construction

10 Multiple Variables

Content Under Construction