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

• 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.