A First Course in Analysis

206 Pages·2013·1.146 MB·English

by  Donald Yau

A First Course
In Analysis
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A First Course
In Analysis
Donald Yau
The Ohio State University at Newark, USA
World Scientific
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A FIRST COURSE IN ANALYSIS
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic
or mechanical, including photocopying, recording or any information storage and retrieval system now known
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the publisher.
ISBN 978-981-4417-85-3
Printed in Singapore.
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To Eun Soo and Hye-Min
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Preface
This is an introductory text on real analysis for undergraduate students. The first
courseinrealanalysisisfullofchallenges,bothfortheinstructorsandthestudents.
Many mathematics majors consider real analysis a difficult course. The transition
from mechanical computation to formal, rigorous proofs is difficult even for many
mathematics majors. Most students beginning a course in real analysis have never
been asked to understand and construct proofs before. Moreover, even if one has
some ideas about how a proof should go, writing it down in a logical manner is a
challenge in itself. This book is written with these challenges in mind.
The prerequisite for this book is a solid background in freshman calculus in one
variable. The intended audience of this book includes undergraduate mathematics
majors and students from other disciplines whose use real analysis. Since this book
is aimed at students who do not have much prior experience with proofs, the pace
is slower in earlier chapters than in later chapters.
In most instances, motivations for new concepts are explained before the actual
definitions. For many concepts that have negations (for example, convergence of
a sequence), such negations are also stated explicitly. Wherever appropriate we
discuss the basic ideas that lead to a proof before the actual proof is given. Such
discussion is intended to help students develop an intuition as to how proofs are
constructed. There are exercises at the end of each section and of each chapter.
Occasionally, some further topics are explored in these additional exercises.
To the Students
Thereareafewthingsthatyoushouldkeepinmindasyouworkthroughthisbook.
A professor of mine, who I shall not name here, once told me this: Nobody teaches
you mathematics. You teach yourself mathematics. You cannot hope to master
the materials in this book simply by watching your instructor lecturing. In fact, if
you could do that, this book is not for you. I have tried to make the materials as
accessible as possible, but you have to do most of the work. You should attempt as
many exercises as possible, whether they are assigned homework or not. Expect to
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viii A First Course in Analysis
do lots of scratch work as you attempt the exercises. After you have written down
a solution to an exercise, read it again and again, and then some more, to see if
every step is logical.
Expect to think deep and hard as you go through this book. When you read
a proof or an example, make sure you understand where each hypothesis is used.
Make sure that you understand every single step in a proof. If you get stuck at a
certain proof or step, let some time elapse and go back to it later. In particular,
do not expect to understand everything the first time you read it. There are many
parts in this book that you should read and think through multiple times if you
want to master them.
Besides this book, the book [Gelbaum and Olmsted (1964)] is highly recom-
mended as a source of many good and exotic examples.
Donald Yau
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Contents
Preface vii
1. Sets, Functions, and Real Numbers 1
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Mathematical Induction. . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Countability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2. Sequences 29
2.1 Sequences of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Properties of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . 44
2.4 Limit Superior and Limit Inferior . . . . . . . . . . . . . . . . . . . . 51
2.5 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3. Series 61
3.1 Convergence of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Alternating Series Test . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Rearrangement of Series . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4. Continuous Functions 83
4.1 Limit Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4 Extreme and Intermediate Value Theorems . . . . . . . . . . . . . . 93
ix