Introduction to Real Analysis (2nd Ed.) by Stoll, M.

Stoll, M. (2000) Introduction to Real Analysis. 2nd Ed. Pearson

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

 

Purpose

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

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

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.

 

 

 

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