- 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.