Basic Analysis I

Basic Analysis I

Introduction to Real Analysis, Volume I

by Ji? Lebl July 11, 2023 (version 6.0)

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Typeset in LATEX. Copyright ?2009?2023 Ji? Lebl

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Contents

Introduction

5

0.1 About this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.2 About analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.3 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1 Real Numbers

23

1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 The set of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Absolute value and bounded functions . . . . . . . . . . . . . . . . . . . . . 36

1.4 Intervals and the size of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.5 Decimal representation of the reals . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Sequences and Series

51

2.1 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2 Facts about limits of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 Limit superior, limit inferior, and Bolzano?Weierstrass . . . . . . . . . . . . . 73

2.4 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.6 More on series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3 Continuous Functions

113

3.1 Limits of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.2 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.3 Extreme and intermediate value theorems . . . . . . . . . . . . . . . . . . . . 130

3.4 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.5 Limits at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.6 Monotone functions and continuity . . . . . . . . . . . . . . . . . . . . . . . . 149

4 The Derivative

155

4.1 The derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.2 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.3 Taylor's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.4 Inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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CONTENTS

5 The Riemann Integral

181

5.1 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.2 Properties of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.3 Fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.4 The logarithm and the exponential . . . . . . . . . . . . . . . . . . . . . . . . 207

5.5 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6 Sequences of Functions

227

6.1 Pointwise and uniform convergence . . . . . . . . . . . . . . . . . . . . . . . 227

6.2 Interchange of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

6.3 Picard's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7 Metric Spaces

255

7.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.2 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

7.3 Sequences and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

7.4 Completeness and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.5 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

7.6 Fixed point theorem and Picard's theorem again . . . . . . . . . . . . . . . . 296

Further Reading

301

Index

303

List of Notation

309

Introduction

0.1 About this book

This first volume is a one semester course in basic analysis. With the second volume it is a year-long course. The book started its life as my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009. I added the metric space chapter to teach Math 521 at University of Wisconsin?Madison (UW). Volume II was added to teach Math 4143/4153 at Oklahoma State University (OSU). A prerequisite for these courses is usually a basic proof course, using for example [H], [F], or [DW].

It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school (such as UIUC 444), but also as a more advanced one-semester course that also covers topics such as metric spaces (such as UW 521). Here are suggestions for a semester course. A slower course such as UIUC 444:

?0.3, ?1.1??1.4, ?2.1??2.5, ?3.1??3.4, ?4.1??4.2, ?5.1??5.3, ?6.1??6.3

A more rigorous course covering metric spaces that runs quite a bit faster (e.g., UW 521):

?0.3, ?1.1??1.4, ?2.1??2.5, ?3.1??3.4, ?4.1??4.2, ?5.1??5.3, ?6.1??6.2, ?7.1??7.6

It should also be possible to run a faster course without metric spaces covering all sections of chapters 0 through 6. The approximate number of lectures given in the section notes through chapter 6 are a very rough estimate and were designed for the slower course. The first few chapters of the book can be used in an introductory proofs course as is done, for example, at Iowa State University Math 201, where this book is used in conjunction with Hammack's Book of Proof [H].

With volume II, one can run a year-long course that covers multivariable topics. In this scenario, it may make sense to cover most of the first volume in the first semester while leaving metric spaces for the beginning of the second semester.

The structure of the beginning of volume I somewhat follows the standard syllabus of UIUC Math 444 and therefore has some similarities with Bartle and Sherbert, Introduction to Real Analysis [BS], which is the standard book at UIUC. A major difference is that we define the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is far more appropriate for a course of this level.

Our approach allows us to fit a course such as UIUC 444 within a semester and still spend some time on the interchange of limits and end with Picard's theorem on the

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