Table Of ContentSpringer Undergraduate Mathematics Series
Advisory Board
P.I. Cameron Queen Mary and Westfield College
M.A.I. Chaplain University ofDundee
K. Erdmann Oxford University
L.C.G. Rogers UniversityofCambridge
E. Siili Oxford University
I.F. Toland UniversityofBath
Other books in this series
A First Course in Discrete Mathematics 1. Anderson
Analytic Methods for Partial Differential Equations G. Evans, ]. Blackledge, P. Yard1ey
Applied Geometry for Computer Graphics and CAD, Second Edition D. Marsh
Basie Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson
Basie Stochastic Processes Z. Brzeiniak and T. Zastawniak
Complex Analysis ]M. Howie
Elementary Differential Geometry A. Press1ey
Elementary Number Theory G.A. ]ones and ].M. ]ones
Elements of Abstract Analysis M. 6 Searc6id
Elements of Logic via Numbers and Sets D.L. ]ohnson
Essential Mathematical Biology N.F. Britton
Essential Topology M.D. Cross1ey
Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker
Further Linear Algebra T.S. Blyth and E.F. Robertson
Geometry R. Fenn
Groups, Rings and Fields D.A.R. Wallace
Hyperbolic Geometry ]. W. Anderson
Information and Coding Theory G.A. ]ones and ].M. ]ones
Introduction to Laplace Transforms and Fourier Series P.P.G. Dyke
Introduction to Ring Theory P.M. Cohn
Introductory Mathematics: Algebra and Analysis G. Smith
Linear Functional Analysis B.P. Rynne and M.A. Youngson
Mathematics for Finance: An Introduction to Financial Engineering M. Capinksi and
T. Zastawniak
Matrix Groups: An Introduction to Lie Group Theory A. Baker
Measure, Integral and Probability, Second Edition M. Capitiksi and E. Kopp
Multivariate Calculus and Geometry, Second Edition S. Dineen
Numerical Methods for Partial Differential Equations G. Evans, ]. Blackledge, P. Yard1ey
Probability Models ].Haigh
Real Analysis ]M. Howie
Sets, Logic and Categories P. Cameron
Special Relativity NM.]. Woodhouse
Symmetries D.L. ]ohnson
Topics in Group Theory G. Smith and O. Tabachnikova
Vector Calculus P.e. Matthews
John M. Howie
Real Analysis
With 35 Figures
, Springer
John M. Howie, CBE, MA, D.Phil, DSc, Hon D.Univ, FRSE
School of Mathematics and Statistics, Mathematical Institute,
University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
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USA. Tel: (206) 432 -78551'1x (206) 432 -7832 emaiJ: ~ URL:www.optech.cum
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TELOS: ISBN 0-387-14222-3, German ediIioa by Birl<baIllOr.ISBN }'7643-510D-4.
Mathemalial in Education anei Raean:h Voi 4 Iuue 3 1995 artide by Ric:hanI J GayIord anei Kazume Nishidare"ralIic: J!naineerinB with CeIIular
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Sprading' JlIII! 33 fig 1. Malhemalial in Edw:ation anei Raean:h Voi 5 '-2 1\J915 artide by Joe BuhIor anei Stan W..., 'SecretI al the
Madelq ConItant' JlIII! 50 fig 1.
British Library Cataloguing in Publication Data
Howie, 1000 Mackintosh
Real ana1ysis. - (Springer undergraduate mathematics
series)
1. Mathematica1 Analysis
1. Tîtle
SIS
ISBN 978-1-85233-314-0
Library of Congress Cata1oging-in-Publication Data
Howie, 1000 M. 0000 Mackintosh)
Real ana1ysis I 1000 M. Howie
p. an. - (Springer undergraduate mathematics series, ISSN 1615-2085)
Indudes bib1iographical references and index.
ISBN 978-1-85233-314-0 ISBN 978-1-4471-0341-7 (eBook)
DOI 10.1007/978-1-4471-0341-7
1. Mathematica1 ana1ysia. 1. Tide. II. Series.
QA300.H694 2001
S15-dc21 00-069839
Apart from any fair deaIing for the purposea of researcb or private study, or aiticism or review, as permitted
under the Copyright, DesigDs and Patents Act 1988, thia publication may only be reproduced, atored or
transmitted, in any fonn or by any means, with the prior pmniasion in writing ofthe publishm, or in the case of
reprographic reproduction in accordance with the terms of Iicences iasued by the Copyright Licensing Agency.
Enquiries conceming reproduction outside thase terms should be sent to the publishers.
Springer Undergraduate Mathematics Series ISSN 1615-2085
ISBN 978-1-85233-314-0
springeronline.com
o Springer-VeriagLondon 2001
OriginaIly published by Springer-Ve rIag London Limited in 2001
The use of registered names, trademarks etc. in thia pub1ication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant Jaws and regu1ations and therefore free for general use.
The publisher maltes no representation, express or implied, with regard to the accuracy of the information
contained in thia book and cannot accept any legal responsibility or liability for any errors or omiasîons that may
bemade.
Typesetting: Camera ready by the author
1213830-5432 Printed on acid-free paper SPIN 11338116
To my grandchildren
Catriona, Sarah, Karen and Fiona,
who may some day want to read this book
Preface
From the point ofview ofstrict logic, a rigorous course on real analysis should
precedea courseoncalculus. Strict logic, is, however,overruledby both history
and practicality. Historically,calculus, withitsoriginsinthe 17thcentury,came
first, and made rapid progress on the basis of informal intuition. Not until
well through the 19th century was it possible to claim that the edifice was
constructed on sound logical foundations. As for practicality, every university
teacher knows that students are not ready for even a semi-rigorous course on
analysis until they have acquired the intuitions and the sheer technical skills
that come from a traditional calculus course.
Real analysis, I have always thought, is the pons asinorv.m1 of modern
mathematics. This shows, I suppose, how much progress we have made in two
thousand years, for it is a great deal more sophisticated than the Theorem of
Pythagoras, which once received that title. All who have taught the subject
know how patient one has to be, for the ideas take root gradually, even in
students of good ability. This is not too surprising, since it took more than
two centuries for calculus to evolve into what we now call analysis, and even a
gifted student, guided by an expert teacher, cannot be expected to grasp all of
the issues immediately.
I have not set out to do anything very original, since in a field as well
establishedasrealanalysisoriginalitytooeasilybecomeseccentricity. Although
it is important to demonstrate the limitations of a visual, intuitive approach
by means of some "strange" examples of functions, too much emphasis on
"pathology" gives altogether the wrong impression ofwhat analysis is about. I
hope that I have avoided that error.
It is, of course, possible to handle many of the fundamental ideas of real
analysiswithin themoregeneralframeworkofmetricspaces. Assuredlythis has
advantages, but I have preferred to take the moreconcrete approach, believing
The bridge of asseSj that is, the bridge between elementary mathematics, which
1
asses can understand, into higher regions ofthought.
VIII Real Analysis
as I do that to add abstraction to the difficulties facing the student is to make
the subject unnecessarily daunting. By the same token, though many of the
notions apply to both real and complex numbers, it seemed that there was
little to gained by drawing attention to this aspect.
My experience suggests that the central notion of a limit is more easily
approached in the context of sequences, and I take this as a starting point.
Sequences, and the closely related topic ofseries, occupy Chapter 2. Chapter
3 (Continuity), Chapter 4 (Differentiation) and Chapter 5 (The Riemann In
tegral) form the core of the book. Chapter 6 introduces the logarithmic and
exponential functions, and the crucial concept ofuniform convergence is intro
duced in Chapter 7 (Sequences and series offunctions).
Any authorsettingout to write a bookon real analysis has to decide where
to introduce the circular functions sin and cos. Logically one ought to delay
their introduction until it can be done "properly", and this is what I would do
if I ever wished (perish the thought!) to write a rigorous GOUT'S d'Analyse in
the grand French manner. Theremust, however, be few students setting out on
a first course in analysis who are unfamiliar with sines and cosines, and I am
certainly not the first author to adopt the practical policy of regarding these
functions as provisionally "known", pending proper definitions. The proper
definitions are given in Chapter 8.
It is all too easy to present mathematics as a set of truths inscribed on
tablets ofstone, complete and perfect. By variousfootnotes and index entries I
havesoughttoemphasisethat thesubject wascreatedby real people. Muchin
terestinginformationonthesepeoplecanbefound intheSt AndrewsHistoryof
Mathematics archive (http://www-history.mes.st-and.ae.uk/history/).
The book contains many worked examples, as well as 190 exercises, for
which briefsolutions are provided at the end ofthe book.
I retired in 1997 after 27 years in the University of St Andrews. It is a
pleasure to record thanks to the University, and in particular to the School of
Mathematicsand Statistics, for their generosityin continuingto give meaccess
to a desk and to the various facilities ofthe Mathematical Institute.
I am grateful to my colleague John O'Connor for his help in creating the
diagrams. Warmest thanks aredue alsoto mycolleaguesKenneth Falconerand
Lars Olsen,whosecommentsonthemanuscripthave, Ihope,eliminatedserious
errors. The responsibility for any imperfections that remain is mine alone.
John M. Howie
University ofSt Andrews
November, 2000
Contents
1. Introductory Ideas. ......................................... 1
1.1 Foreword for the Student: Is Analysis Necessary? ............. 1
1.2 The Concept of Number. .................................. 3
1.3 The Language ofSet Theory ............................... 4
1.4 Real Numbers. ........................................... 7
1.5 Induction................................................ 12
1.6 Inequalities.............................................. 18
2. Sequences and Series 27
2.1 Sequences............................................... 27
2.2 Sums, Products and Quotients 33
2.3 Monotonic Sequences 37
2.4 Cauchy Sequences ........................................ 42
2.5 Series................................................... 47
2.6 The Comparison Test 50
2.7 Series ofPositive and Negative Terms 58
3. Functions and Continuity. .................................. 63
3.1 Functions, Graphs 63
3.2 Sums, Products, Compositions; Polynomialand Rational Func-
tions 66
3.3 Circular Functions. ....................................... 70
3.4 Limits.................................................. 73
3.5 Continuity............................................... 81
3.6 Uniform Continuity 90
3.7 Inverse Functions. ........................................ 94
x Contents
4. Differentiation.............................................. 99
4.1 The Derivative. .......................................... 99
4.2 The Mean Value Theorems 105
4.3 Inverse Functions 110
4.4 Higher Derivatives 113
4.5 Taylor's Theorem 116
5. Integration 119
5.1 The Riemann Integral. 119
5.2 Classes ofIntegrable Functions 126
5.3 Properties ofIntegrals 131
5.4 The Fundamental Theorem 138
5.5 Techniques ofIntegration 143
5.6 Improper Integrals ofthe First Kind 150
5.7 Improper Integrals ofthe Second Kind 158
6. The Logarithmic and Exponential Functions 165
6.1 A Function Defined by an Integral 165
6.2 The Inverse Function 168
6.3 Further Propertiesofthe Exponential and LogarithmicFunctions176
7. Sequences and Series ofFunctions 181
7.1 Uniform Convergence 181
7.2 Uniform Convergence ofSeries 192
7.3 Power Series 201
8. The Circular Functions 217
8.1 Definitions and Elementary Properties 217
8.2 Length 220
9. Miscellaneous Examples 229
9.1 Wallis's Formula 229
9.2 Stirling's Formula 230
9.3 A Continuous, Nowhere Differentiable Function 234
Solutions to Exercises 237
The Greek Alphabet 269
Bibliography 271
Index 273
1
Introductory Ideas
1.1 Foreword for the Student: Is Analysis
Necessary?
In writing this book my assumption has been that you have encountered the
fundamental ideas of analysis (function, limit, continuity, differentiation, inte
gration) in a standard course on calculus. For many purposes there is no harm
at all in an informal approach, in which a continuous function is one whose
graph has no jumps and a differentiable function is one whose graph has no
sharp corners, and in which it is "obvious" that (say) the sum oftwo or more
continuous functions is continuous. On the other hand, it is not obvious from
the graph that the function f defined by
={
f(x) Xsin(lfx) ifxi- 0
o =
ifx 0
=
is continuous but not differentiable at x 0, since the function takes the value
o
infinitely often in any interval containing0, and so it is not really possible to
drawthegraphproperly.Moresignificantly,sincethesinefunction iscontinuous
it might seem obvious that the function S defined by
S(x) =sinx _ sin2x + sin3x _ ...
1 2 3
J. M. Howie, Real Analysis
© Springer-Verlag London Limited 2001
