Introduction to Real Analysis M361K

Introduction to Real Analysis M361K

Last Updated: July 2011

Preface

These notes are for the basic real analysis class, M361K (The more advanced class is M365C.) They were written, used, revised and revised again and again over the past decade. Contributors to the text include both TA's and instructors: Grant Lakeland, Cody Patterson, Alistair Windsor, Tim Blass, David Paige, Louiza Fouli, Cristina Caputo and Ted Odell.

The subject is calculus on the real line, done rigorously. The main topics are sequences, limits, continuity, the derivative and the Riemann integral. It is a challenge to choose the proper amount of preliminary material before starting with the main topics. In early editions we had too much and decided to move some things into an appendix to Chapter 2 (at the end of the notes) and to let the instructor choose what to cover. We also removed much of the topology on R material from Chapter 3 and put it in an appendix. In a one semester course we are usually able to do the majority of problems from Chapter 3?6 and a small selection of certain preliminary problems from Chapter 2 and the two appendices.

July 2011

Contents

Chapter 1. Introduction

1

1. Goals

1

2. Proofs

1

3. Logic

2

Chapter 2. Preliminaries: Numbers and Functions

5

1. Functions

8

2. The Absolute Value

13

3. Intervals

15

Chapter 3. Sequences

19

1. Limits and the Archimedean Property

19

2. Properties of Convergence

25

3. Monotone Sequences

28

4. Subsequences

33

5. Cauchy Sequences

34

6. Decimals

35

7. Supremums and Completeness

39

8. Real and Rational Exponents

43

Chapter 4. Limits of Functions and Continuity

47

1. Limits of Functions

47

2. Continuous Functions

55

3. Theorems About Continuous Functions

59

4. Uniform Continuity

62

Chapter 5. Differentiation

65

1. Derivatives

65

2. Theorems About Differentiable Functions

68

Chapter 6. Integration

71

1. The Definition

71

2. Integrable Functions

77

3. Properties of Integrals

78

4. Fundamental Theorems of Calculus

80

5. Integration Rules

81

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CONTENTS

Appendix A. Prerequisite Knowledge

83

1. Set Theory

83

2. The Field Properties of the Real Numbers

85

3. The Order Properties of the Real Numbers

88

4. The Ordered Field Properties of the Real Numbers

90

5. Mathematical Induction

91

Appendix B. Appendix to Chapter 3

93

1. Cardinality

93

2. Open and Closed Sets

96

3. Compactness

99

4. Sequential Limits and Closed Sets

100

CHAPTER 1

Introduction

1. Goals

The purpose of this course is three-fold:

(1) to provide an introduction to the basic definitions and theorems of calculus and real analysis.

(2) to provide an introduction to writing and discovering proofs of mathematical theorems. These proofs will go beyond the mechanical proofs found in your Discrete Mathematics course.

(3) and most importantly to let you experience the joy of mathematics: the joy of personal discovery.

2. Proofs

Hopefully all of you have seen some proofs before. A proof is the name that mathematicians give to an explanation that leaves no doubt. The level of detail in this explanation depends on the audience for the proof. Mathematicians often skip steps in proofs and rely on the reader to fill in the missing steps. This can have the advantage of focusing the reader on the new or crucial ideas in the proof but can easily lead to frustration if the reader is unable to fill in the missing steps. More seriously these missing steps can easily conceal mistakes: many mistakes in proofs, particularly at the undergraduate level, begin with "it is obvious that".

In this course we will try to avoid missing any steps in our proofs. Each statement should follow from a previous one by a simple property of arithmetic, by a definition, or by a previous theorem, and this justification should be clearly stated in plain language. Writing clear proofs is a skill in itself. Often the shortest proof is not the clearest.

There is no mechanical process to produce a proof but there are some basic guidelines you should follow. The most basic is that every object that appears should be defined; when a variable, function, or set appears we should be able to look back and find a statement defining that object:

(1) Let > 0 be arbitrary.

1

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