• Title: Introduction to Real Analysis
• Edition: 2nd
• Author: Manfred Stoll
• Publisher: Pearson

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Purpose

• Learn?and summarize contents in the book in my words.

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Contents

• 1 The Real Number System
  • 1.1 Sets and Operations on Sets
  • 1.2 Functions
  • 1.3 Mathematical Induction
  • 1.4 The Least Upper Bound Property
  • 1.5 Consequences of the Least Upper Bound Property
  • 1.6 Binary and Ternary Expansions
  • 1.7 Countable and Uncountable Sets
• 2 Sequence of Real Numbers
  • 2.1 Convergent Sequences
  • 2.2 Limit Theorems
  • 2.3 Monotone Sequences
  • 2.4 Subsequences and the Bolzano-Weierstrass Theorem
  • 2.5 Limit Superior and Inferior of a Sequence
  • 2.6 Cauchy Sequences
  • 2.7 Series of Real Numbers
• 3 Structure of Point Sets
  • 3.1 Open and Closed Sets
  • 3.2 Compact Sets
  • 3.3 The Cantor Set
• 4 Limit and Continuity
  • 4.1 Limit of a Function
  • 4.2 Continuous Functions
  • 4.3 Uniform Continuity
  • 4.4 Monotone Functions and Discontinuities
• 5 Differentiation
  • 5.1 The Derivative
  • 5.2 The Mean Value Theorem
  • 5.3 L’Hospital’s Rule
  • 5.4 Newton’s Method
• 6 The Riemann and Riemann-Stieltjes Integral
• 7 Series of Real Numbers
• 8 Sequence and Series of Functions
• 9 Orthogonal Functions and Fourier Series
• 10 Lebesgue Measure and Integration

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Summary

1 The Real Number System

1.3 Mathematical Induction

1.3.1 Theorem (Principle of Mathematical Induction)

Precondition

• For each n \in \mathbb{N} let P(n) be a statement about the positive integer n .
• (a) P(n) is true.
• (b) If P(k) is true, P(k+1) is true.

Theorem

If (a) and (b) are true, then P(n) is true for all n \in \mathbb{N}

 

6 The Riemann and Riemann-Stieltjes Integral

• Cauchy proved the fundamental theorem of calculus.
• “The modern definition of integration was developed by in 1853 by Georg Bernhard Riemann (1826-1866).”
• Lebesgue’s theorem seems important.
• The Riemann-Stieltjes integral arises in many applications in both mathematics and physics.
• The Riemann-Stieltjes integral involves only minor modifications in the definition of the Riemann integral.