Real Analysis: A Constructive Approach Through Interval Arithmetic

321 Pages·2019·2.555 MB·English

by  Mark Bridger

Pure and Applied
S ally 38
The UNDERGRADUATE TEXTS
SERIES
Real Analysis
A Constructive
Approach Through
Interval Arithmetic
Mark Bridger
Real Analysis
A Constructive
Approach Through
Interval Arithmetic

T h e UNDERGRADUATE TEXTS • 38
SERIES
Pure and Applied
Sally
Real Analysis
A Constructive
Approach Through
Interval Arithmetic
Mark Bridger
EDITORIAL COMMITTEE
Gerald B. Folland (Chair) Steven J. Miller
Jamie Pommersheim Serge Tabachnikov
Originally published in 2007 by John Wiley & Sons, Inc.
2010 Mathematics Subject Classification. Primary 03F60, 03F55, 97-01, 97Ixx,
26-01, 26Axx.
For additional informationand updates on this book, visit
www.ams.org/bookpages/amstext-38
Library of Congress Cataloging-in-Publication Data
Names: Bridger,Mark,1942-author.
Title: Realanalysis: aconstructiveapproachthroughintervalarithmetic/MarkBridger.
Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: AMS
pure and applied undergraduate texts ; volume 38 | Includes bibliographical references and
index.
Identifiers: LCCN2019006280|ISBN9781470451448(alk. paper)
Subjects: LCSH:Mathematicalanalysis. |Continuity. |Differentiablefunctions. |Intervalanaly-
sis(Mathematics)|AMS:Mathematicallogicandfoundations–Prooftheoryandconstructive
mathematics–Constructiveandrecursiveanalysis. msc|Mathematicallogicandfoundations
–Prooftheoryandconstructive mathematics–Intuitionistic mathematics. msc|Mathemat-
icseducation–Instructionalexposition(textbooks,tutorialpapers,etc.). msc|Mathematics
education – Analysis – Analysis. msc | Real functions – Instructional exposition (textbooks,
tutorial papers, etc.). msc | Real functions – Functions of one variable – Functions of one
variable. msc
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CONTENTS
Preface vii
Acknowledgements xi
Introduction xiii
0 Preliminaries 1
0.1 The Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 The Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 The Real Numbers and Completeness 11
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Families of Intersecting Intervals . . . . . . . . . . . . . . . . . . . . 22
1.3 Fine Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 De(cid:31)nition of the Reals . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5 Real Number Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6 Rational Approximations . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7 Real Intervals and Completeness . . . . . . . . . . . . . . . . . . . . 58
1.8 Limits and Limiting Families . . . . . . . . . . . . . . . . . . . . . . 63
Appendix: The Goldbach Number and Trichotomy . . . . . . . . . . . . . 67
2 An Inverse Function Theorem and Its Application 69
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1 Functions and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2 An Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . 74
2.3 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . 83
2.4 Natural Logs and the Euler Number (cid:104) . . . . . . . . . . . . . . . . . 94
3 Limits, Sequences and Series 99
3.1 Sequences and Convergence . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3 Series of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Appendix I: Some Properties of Exp and Log . . . . . . . . . . . . . . . . 131
Appendix II: Rearrangements of Series . . . . . . . . . . . . . . . . . . . . 134
4 Uniform Continuity 139
4.1 De(cid:31)nitions and Elementary Properties . . . . . . . . . . . . . . . . . 139
4.2 Limits and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Appendix I: Are There Non-Continuous Functions? . . . . . . . . . . . . . 157
Appendix II: Continuity of Double-Sided Inverses . . . . . . . . . . . . . . 161
v
vi CONTENTS
Appendix III: The Goldbach Function . . . . . . . . . . . . . . . . . . . . 163
5 The Riemann Integral 165
5.1 De(cid:31)nition and Existence . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 Extensions and Improper Integrals . . . . . . . . . . . . . . . . . . . 176
6 Di(cid:30)erentiation 185
6.1 De(cid:31)nitions and Basic Properties . . . . . . . . . . . . . . . . . . . . 185
6.2 The Arithmetic of Di(cid:30)erentiability . . . . . . . . . . . . . . . . . . . 191
6.3 Two Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . 196
6.4 Derivative Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.5 Integral Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7 Sequences and Series of Functions 223
7.1 Sequences of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.2 Integrals and Derivatives of Sequences . . . . . . . . . . . . . . . . . 233
7.3 Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.5 The Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 261
Appendix: Raabe’s Test and Binomial Issues . . . . . . . . . . . . . . . . 269
8 The Complex Numbers and Fourier Series 271
8.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
8.1 The Complex Numbers C . . . . . . . . . . . . . . . . . . . . . . . . 275
8.2 Complex Functions and Vectors . . . . . . . . . . . . . . . . . . . . . 278
8.3 Fourier Series Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 284
References 295
Index 297
PREFACE
What is a constructive approach, and why should one take it?
If you look at the table of contents for this book, you’ll see mostly familiar
topics, but with a slightly di(cid:30)erent emphasis. There’s a long chapter on the real
numbers,followedbyoneon“AnInverseFunctionTheorem.”Thechapteronlimits,
sequencesandseriesisfollowedbyoneonuniform continuity–whynotjustpointwise
continuity? AchapterontheRiemannintegralisfollowedbyoneondi(cid:30)erentiation–
but it’s actually uniform di(cid:30)erentiation. All of these departures from the structure
of the usual real analysis text result from a careful reassessment of the role of the
course in the technical education of undergraduates.
Not every student in Real Analysis is a math major, and, in many schools, only
a small percentage of math majors intend to do graduate work in mathematics.
A modern course is populated by a wide range of students. Some are headed for
careers in secondary education, while there is often a large contingent from the
physical sciences and an even larger group from computer science. These students
are in the course because they need or want more than a cookbook calculus course.
Some need to know more about computability and calculability of (cid:32)oating-point
numbers, hence more about the actual nature of the reals. They also need to know
about continuity because they need to know about approximations; some need to
know about convergence and improper integrals because they need to know about
computing special functions and transforms.
But real analysis is not primarily focused on computing. It is, signi(cid:31)cantly, a
course that shapes the way students think about mathematics. Very often it is a
student’s introduction to precise reasoning and writing.
So: I begin with a careful construction of the real numbers, the (cid:31)eld on which
most of analysis is played. The approach here, due to Gabriel Stolzenberg, is via
intervalsofrationalnumbersandthearithmeticofsuchintervals. Themanyelemen-
tarytheorems abouttheproperties of this arithmeticlaterreappearas properties of
the real numbers, and verifying them provides a gentle introduction to the art and
practice of devising and writing readable and correct proofs. Furthermore, there is
a useful metaphor: a rational interval is exactly what is obtained when a scientist
uses instruments of limited (but known) accuracy to measure something. Families
of rational intervals then correspond to multiple measurements, and the condition
on a family that any two of its intervals must meet establishes the consistency of its
measurements. Finally, a real number is de(cid:31)ned to be a family of rational intervals
thatisconsistentinthissense,andthatcontainsintervalsofarbitrarilysmalllength.
Intervalarithmetic,carriedovertofamiliesofintervals,nowbecomesrealarithmetic,
and conditions on the lengths of intervals become the properties of approximation
of reals by rationals.
At this point, the students see that the reals have a far more complex structure
than the rationals. One important example is the traditional Law of Trichotomy,
namely that precisely one of (cid:123)(cid:63)(cid:124), (cid:124) (cid:63)(cid:123), or (cid:123)=(cid:124) must hold. This property holds
vii
viii Preface
for the rationals, since rational arithmetic is basically integer arithmetic. However,
reals can, in general, only be approximated by rationals. Modern computer algebra
systems allow the user to specify a tolerance, which is expressed as the number of
decimalplaces.Thisnumbercanbechosenaslargeasonepleases,butnotin(cid:31)nitely
large. The corresponding tolerance, say (cid:18), tells us how closely we can distinguish
reals using the computer’s rational representation. These considerations lead to the
formulation of real number comparison that we prove and use throughout the book.
(cid:18)-Trichotomy Given any tolerance (cid:18)(cid:65)0, then for any reals (cid:123)and (cid:124), (cid:123)(cid:63)(cid:124), (cid:124) (cid:63)(cid:123),
or (cid:123) and (cid:124) are within (cid:18) of each other.
Thus, a construction ofthereals based on rational measurementand an analysis
of what we can actually calculate produces a concordance of theory and practice
that students of the sciences easily relate to.
Usingthenotionof(cid:18)-trichotomyasatoolforcomparingrealnumbersenablesus
todescribeabisection-likealgorithmfor(cid:31)ndingtheinverseofafunction(cid:105),providing
it satis(cid:31)es upper and lower bounds on its di(cid:30)erence quotient (cid:105)((cid:124)(cid:124))(cid:3)(cid:105)(cid:123)((cid:123)). This leads
directly to the construction of (cid:113)th root, exponential, and logarithm(cid:3) functions.
Another hallmark of the constructivist program is its emphasis on uniform vs.
pointwise continuity:
(cid:105) is pointwise continuous at (cid:100) if, given any (cid:18)(cid:65)0 we can (cid:31)nd a (cid:17) ((cid:18))(cid:65)0 such
(cid:100)
•
that (cid:105)((cid:123)) (cid:105)((cid:100)) (cid:63)(cid:18) whenever (cid:123) (cid:100) (cid:63)(cid:17) ((cid:18)).
(cid:100)
| (cid:3) | | (cid:3) |
(cid:105) is uniformly continuous on if, given any (cid:18)(cid:65)0 we can (cid:31)nd a (cid:17)((cid:18))(cid:65)0 such
• S
that (cid:105)((cid:123)) (cid:105)((cid:124)) (cid:63)(cid:18) whenever (cid:123) (cid:124) (cid:63)(cid:17)((cid:18)) and (cid:123)(cid:62)(cid:124) .
| (cid:3) | | (cid:3) | (cid:53)S
Uniform continuity on implies pointwise continuity at each point of , but the
S S
converse is not true: there is no general procedure for constructing a single (cid:17) from
the in(cid:31)nitely many (cid:17) . Not only is uniform continuity a stronger notion, it is the
(cid:100)
more desirable version of continuity since it is the one most useful in studying con-
vergenceandintegrability. Itturnsoutthattheusualproofsthatthebasicfunctions
of analysis are pointwise continuous also prove that these functions are uniformly
continuous on appropriate intervals. We exploit this fact from the very beginning
and only use the stronger and more important uniform version of continuity.
We take a similar approach to di(cid:30)erentiability. Instead of talking about the
derivative of a function at a point, we talk about the derivative function on an
interval. As with uniform continuity, this notion of uniform di(cid:30)erentiability is the
one that is of most importance in later theory and applications. In fact, it is an
approach that generalizes readily to vector-valued functions of several variables.
An important consequence of using uniform notions is that they produce trans-
parent proofs of important theorems such as the existence of the Riemann integral
and the Fundamental Theorem of Calculus.
The pointwise versions of continuity and di(cid:30)erentiability do lead to a number
of classical examples of functions which are or aren’t continuous or di(cid:30)erentiable
on various dense or nowhere dense subsets of intervals. Since we are emphasizing
uniform notions, these examples are relegated to discussions in an appendix and a
few exercises, which can be covered at the discretion of the instructor.
Preface ix
In summary, then, this is neither a text in numerical analysis nor one intended
solely to prepare students to be professional mathematicians. It is a thoroughly
rigorous modern account of the theoretical underpinnings of calculus; and, being
constructive in nature, every proof of every result is direct and ultimately computa-
tionallyveri(cid:31)able(atleast in principle). In particular, existence is neverestablished
by showing that the assumption of non-existence leads to a contradiction. By look-
ing through the index or table of contents, you’ll see that nothing of importance for
undergraduateshasbeenlefto(cid:30)orcompromisedbyourapproach. Thepayo(cid:30)ofthe
constructive approach, however, is that it makes sense–not just to math majors,
but to students from all branches of the sciences.