Introduction to Real Analysis

• Italic: Definition
• Bold: Theorem/Corollary

Content

Chapter 1

• 1.1 Sets and Functions
  • Set Equality
  • Set Union
  • Set Intersection
  • Set Complement
  • Empty Set
  • Disjoint Sets

Chapter 2

• 2.1 The Algebraic and Order Properties of R

  • Field Axioms
  • Uniqueness of Identity and Multiplication by 0
  • Uniqueness of Reciprocal and Multiplication Resulting in 0
  • Natural Numbers
  • Integers
  • Rational Numbers
  • Irrational Numbers
  • Square Root of 2 is Irrational
  • Order Properties of Real Numbers
  • Some Inequality Notations
  • Inference from Inequalities
  • Various Order Theorems
  • Proposition: No “Smallest” Positive Real Numbers Can Exist
  • Notion of Limit from Order
  • Inference from Multiplication and Inequality

• 2.2 Absolute Value and the Real Line

  • Absolute Value
  • Additional Property of Absolute Values
  • Triangle Inequality
  • Neighborhood
  • Uniqueness of a Point in Neighborhoods

• 2.3 The Completeness Property of R

  • Bounded Set
  • Supremum and Infimum
  • Completeness Property of Real Numbers

• 2.4 Applications of the Supremum Property

• 2.5 Intervals

Chapter 3

• 3.1 Sequences and Their Limits
  • Sequence of Real Numbers
• 3.2 Limit Theorems
• 3.3 Monotone Sequences
  • Monotone Sequence
• 3.4 Subsequences and the Bolzano-Weierstrass Theorem
• 3.5 The Cauchy Criterion
• 3.6 Properly Divergent Sequences
• Various Methods for Proving a Sequence of Real Numbers is Convergent

Chapter 4

• 4.1 Limits of Function
• 4.2 Limit Theorems
• 4.3 Some Extensions of the Limit Concept

Chapter 5

• 5.1 Continuous Functions
• 5.2 Combinations of Continuous Functions
• 5.3 Continuous Functions on Intervals