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A Companion to Analysis
A Second First and First Second Course in Analysis


T.W.K¨rner
o
Trinity Hall
Cambridge


Note This is the ¬rst draft for a possible book. I would therefore be glad to receive
corrections at twk@dpmms.cam.ac.uk. Senders of substantial lists of errors or of lists of
substantial errors will receive small rewards and large thanks. General comments are also
welcome. Please refer to this as DRAFT F3 (note that Appendix K was reordered between
drafts E and F ). Please do not say ˜I am sure someone else has noticed this™ or ˜This
is too minor to matter™. Everybody notices di¬erent things and no error is too small to
confuse somebody.
ii A COMPANION TO ANALYSIS

[Archimedes] concentrated his ambition exclusively upon those specula-
tions which are untainted by the claims of necessity. These studies, he be-
lieved, are incomparably superior to any others, since here the grandeur and
beauty of the subject matter vie for our admiration with the cogency and
precision of the methods of proof. Certainly in the whole science of geome-
try it is impossible to ¬nd more di¬cult and intricate problems handled in
simpler and purer terms than in his works. Some writers attribute it to his
natural genius. Others maintain that phenomenal industry lay behind the
apparently e¬ortless ease with which he obtained his results. The fact is that
no amount of mental e¬ort of his own would enable a man to hit upon the
proof of one of Archimedes™ theorems, and yet as soon as it is explained to
him, he feels he might have discovered it himself, so smooth and rapid is the
path by which he leads us to the required conclusion.
Plutarch Life of Marcellus [Scott-Kilvert™s translation]


It may be observed of mathematicians that they only meddle with such
things as are certain, passing by those that are doubtful and unknown. They
profess not to know all things, neither do they a¬ect to speak of all things.
What they know to be true, and can make good by invincible argument, that
they publish and insert among their theorems. Of other things they are silent
and pass no judgment at all, choosing rather to acknowledge their ignorance,
than a¬rm anything rashly.
Barrow Mathematical Lectures


For [A. N.] Kolmogorov mathematics always remained in part a sport.
But when . . . I compared him with a mountain climber who made ¬rst as-
cents, contrasting him with I. M. Gel´fand whose work I compared with the
building of highways, both men were o¬ended. ˜ . . . Why, you don™t think
I am capable of creating general theories?™ said Andre˜ Nikolaevich. ˜Why,
±
you think I can™t solve di¬cult problems?™ added I. M.
V. I. Arnol´d in Kolmogorov in Perspective
Contents

Introduction vii

1 The real line 1
1.1 Why do we bother? . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 The fundamental axiom . . . . . . . . . . . . . . . . . . . . . 9
1.5 The axiom of Archimedes . . . . . . . . . . . . . . . . . . . . 10
1.6 Lion hunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 The mean value inequality . . . . . . . . . . . . . . . . . . . . 18
1.8 Full circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9 Are the real numbers unique? . . . . . . . . . . . . . . . . . . 23

2 A ¬rst philosophical interlude ™™ 25
2.1 Is the intermediate value theorem obvious? ™™ . . . . . . . . 25

3 Other versions of the fundamental axiom 31
3.1 The supremum . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Bolzano-Weierstrass theorem . . . . . . . . . . . . . . . . 37
3.3 Some general remarks . . . . . . . . . . . . . . . . . . . . . . . 42

4 Higher dimensions 43
4.1 Bolzano-Weierstrass in higher dimensions . . . . . . . . . . . . 43
4.2 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 A central theorem of analysis . . . . . . . . . . . . . . . . . . 56
4.4 The mean value theorem . . . . . . . . . . . . . . . . . . . . . 60
4.5 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 The general principle of convergence . . . . . . . . . . . . . . 66

5 Sums and suchlike ™ 75
5.1 Comparison tests ™ . . . . . . . . . . . . . . . . . . . . . . . . 75

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iv A COMPANION TO ANALYSIS

Conditional convergence ™ . . . . . . .
5.2 . . . . . . . . . . . . . 78
Interchanging limits ™ . . . . . . . . .
5.3 . . . . . . . . . . . . . 83
The exponential function ™ . . . . . .
5.4 . . . . . . . . . . . . . 91
The trigonometric functions ™ . . . . .
5.5 . . . . . . . . . . . . . 98
The logarithm ™ . . . . . . . . . . . .
5.6 . . . . . . . . . . . . . 102
Powers ™ . . . . . . . . . . . . . . . .
5.7 . . . . . . . . . . . . . 109
The fundamental theorem of algebra ™
5.8 . . . . . . . . . . . . . 113

6 Di¬erentiation 121
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 The operator norm and the chain rule . . . . . . . . . . . . . . 127
6.3 The mean value inequality in higher dimensions . . . . . . . . 136

7 Local Taylor theorems 141
7.1 Some one dimensional Taylor theorems . . . . . . . . . . . . . 141
7.2 Some many dimensional local Taylor theorems . . . . . . . . . 146
7.3 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8 The Riemann integral 169
8.1 Where is the problem ? . . . . . . . . . . . . . . . . . . . . . . 169
8.2 Riemann integration . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Integrals of continuous functions . . . . . . . . . . . . . . . . . 182
First steps in the calculus of variations ™
8.4 . . . . . . . . . . . . 190
8.5 Vector-valued integrals . . . . . . . . . . . . . . . . . . . . . . 202

9 Developments and limitations ™ 205
9.1 Why go further? . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.2 Improper integrals ™ . . . . . . . . . . . . . . . . . . . . . . . 207
9.3 Integrals over areas ™ . . . . . . . . . . . . . . . . . . . . . . 212
9.4 The Riemann-Stieltjes integral ™ . . . . . . . . . . . . . . . . 217
9.5 How long is a piece of string? ™ . . . . . . . . . . . . . . . . . 224

10 Metric spaces 233
10.1 Sphere packing ™ . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2 Shannon™s theorem ™ . . . . . . . . . . . . . . . . . . . . . . . 236
10.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.4 Norms, algebra and analysis . . . . . . . . . . . . . . . . . . . 246
10.5 Geodesics ™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
v
Please send corrections however trivial to twk@dpmms.cam.ac.uk

11 Complete metric spaces 263
11.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
11.2 The Bolzano-Weierstrass property . . . . . . . . . . . . . . . . 272
11.3 The uniform norm . . . . . . . . . . . . . . . . . . . . . . . . 275
11.4 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . 279
11.5 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
11.6 Fourier series ™ . . . . . . . . . . . . . . . . . . . . . . . . . . 298

12 Contractions and di¬erential equations 303
12.1 Banach™s contraction mapping theorem . . . . . . . . . . . . . 303
12.2 Solutions of di¬erential equations . . . . . . . . . . . . . . . . 305
12.3 Local to global ™ . . . . . . . . . . . . . . . . . . . . . . . . . 310
12.4 Green™s function solutions ™ . . . . . . . . . . . . . . . . . . . 318

13 Inverse and implicit functions 329
13.1 The inverse function theorem . . . . . . . . . . . . . . . . . . 329
13.2 The implicit function theorem ™ . . . . . . . . . . . . . . . . . 339
13.3 Lagrange multipliers ™ . . . . . . . . . . . . . . . . . . . . . . 347

14 Completion 355
14.1 What is the correct question? . . . . . . . . . . . . . . . . . . 355
14.2 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
14.3 Why do we construct the reals? ™ . . . . . . . . . . . . . . . . 364
14.4 How do we construct the reals? ™ . . . . . . . . . . . . . . . . 369
14.5 Paradise lost? ™™ . . . . . . . . . . . . . . . . . . . . . . . . 375

A The axioms for the real numbers 379

B Countability 383

C On counterexamples 387

D A more general view of limits 395

E Traditional partial derivatives 401

F Inverse functions done otherwise 407

G Completing ordered ¬elds 411

H Constructive analysis 415

I Miscellany 421
vi A COMPANION TO ANALYSIS

J Executive summary 427

K Exercises 431

Bibliography 603

Index 607
Introduction

In his autobiography [12], Winston Churchill remembered his struggles with
Latin at school. ˜ . . . even as a schoolboy I questioned the aptness of the
Classics for the prime structure of our education. So they told me how Mr
Gladstone read Homer for fun, which I thought served him right.™ ˜Naturally™
he says ˜I am in favour of boys learning English. I would make them all learn
English; and then I would let the clever ones learn Latin as an honour, and
Greek as a treat.™
This book is intended for those students who might ¬nd rigorous analysis
a treat. The content of this book is summarised in Appendix J and corre-
sponds more or less (more rather than less) to a recap at a higher level of the
¬rst course in analysis followed by the second course in analysis at Cambridge
in 2003 together with some material from various methods courses (and thus
corresponds to about 60 to 70 hours of lectures). Like those courses, it aims
to provide a foundation for later courses in functional analysis, di¬erential
geometry and measure theory. Like those courses also, it assumes comple-
mentary courses such as those in mathematical methods and in elementary
probability to show the practical uses of calculus and strengthen computa-
tional and manipulative skills. In theory, it starts more or less from scratch
but the reader who ¬nds the discussion of section 1.1 ba¬„ing or the , δ
arguments of section 1.2 novel will probably ¬nd this book unrewarding.
This book is about mathematics for its own sake. It is a guided tour of a
great but empty Opera House. The guide is enthusiastic but interested only
in sight-lines, acoustics, lighting and stage machinery. If you wish to see the
stage ¬lled with spectacle and the air ¬lled with music you must come at
another time and with a di¬erent guide.
Although I hope this book may be useful to others, I wrote it for stu-
dents to read either before or after attending the appropriate lectures. For
this reason, I have tried to move as rapidly as possible to the points of dif-
¬culty, show why they are points of di¬culty and explain clearly how they
are overcome. If you understand the hardest part of a course then, almost
automatically, you will understand the easiest. The converse is not true.

vii
viii A COMPANION TO ANALYSIS

In order to concentrate on the main matter in hand, some of the sim-
pler arguments have been relegated to exercises. The student reading this
book before taking the appropriate course may take these results on trust
and concentrate on the central arguments which are given in detail. The
student reading this book after taking the appropriate course should have
no di¬culty with these minor matters and can also concentrate on the cen-
tral arguments. I think that doing at least some of the exercises will help
students to ˜internalise™ the material but I hope that even students who skip
most of the exercises can pro¬t from the rest of the book.
I have included further exercises in Appendix K. Some are standard, some
form commentaries on the main text and others have been taken or adapted
from the Cambridge mathematics exams. None are just ˜makeweights™, they
are all intended to have some point of interest. I have tried to keep to
standard notations but a couple of notational points are mentioned in the
index under the heading notation.
I have not tried to strip the subject down to its bare bones. A skeleton
is meaningless unless one has some idea of the being it supports and that
being in turn gains much of its signi¬cance from its interaction with other
beings, both of its own species and of other species. For this reason, I have
included several sections marked by a ™. These contain material which is
not necessary to the main argument but which sheds light on it. Ideally, the
student should read them but not study them with anything like the same
attention which she devotes to the unmarked sections. There are two sections
marked ™™ which contain some, very simple, philosophical discussion. It is
entirely intentional that removing the appendices and the sections marked
with a ™ more than halves the length of the book.
My ¬rst glimpse of analysis was in Hardy™s Pure Mathematics [23] read
when I was too young to really understand it. I learned elementary analysis
from Ferrar™s A Textbook of Convergence [17] (an excellent book for those
making the transition from school to university, now, unfortunately, out of
print) and Burkill™s A First Course in Mathematical Analysis [10]. The books
of Kolmogorov and Fomin [30] and, particularly, Dieudonn´ [13] showed me
e
that analysis is not a collection of theorems but a single coherent theory.
Stromberg™s book An Introduction to Classical Real Analysis [45] lies perma-
nently on my desk for browsing. The expert will easily be able to trace the
in¬‚uence of these books on the pages that follow. If, in turn, my book gives
any student half the pleasure that the ones just cited gave me, I will feel well
repaid.
Cauchy began the journey that led to the modern analysis course in his
´
lectures at the Ecole Polytechnique in the 1820™s. The times were not propi-
tious. A reactionary government was determined to keep close control over
ix
Please send corrections however trivial to twk@dpmms.cam.ac.uk

the school. The faculty was divided along fault lines of politics, religion and
age whilst physicists, engineers and mathematicians fought over the contents
of the courses. The student body arrived insu¬ciently prepared and then
divided its time between radical politics and worrying about the job market
(grim for both sta¬ and students). Cauchy™s course was not popular1 .
Everybody can sympathise with Cauchy™s students who just wanted to
pass their exams and with his colleagues who just wanted the standard ma-
terial taught in the standard way. Most people neither need nor want to
know about rigorous analysis. But there remains a small group for whom
the ideas and methods of rigorous analysis represent one of the most splen-
did triumphs of the human intellect. We echo Cauchy™s de¬ant preface to his
printed lecture notes.

As to the methods [used here], I have sought to endow them
with all the rigour that is required in geometry and in such a
way that I have not had to have recourse to formal manipula-
tions. Such arguments, although commonly accepted . . . cannot
be considered, it seems to me, as anything other than [sugges-
tive] to be used sometimes in guessing the truth. Such reasons
[moreover] ill agree with the mathematical sciences™ much vaunted
claims of exactitude. It should also be observed that they tend to
attribute an inde¬nite extent to algebraic formulas when, in fact,
these formulas hold under certain conditions and for only certain
values of the variables involved. In determining these conditions
and these values and in settling in a precise manner the sense of
the notation and the symbols I use, I eliminate all uncertainty.
. . . It is true that in order to remain faithful to these principles,
I sometimes ¬nd myself forced to depend on several propositions
that perhaps seem a little hard on ¬rst encounter . . . . But, those
who will read them will ¬nd, I hope, that such propositions, im-
plying the pleasant necessity of endowing the theorems with a
greater degree of precision and restricting statements which have
become too broadly extended, will actually bene¬t analysis and
will also provide a number of topics for research, which are surely
not without importance.




1
Belhoste™s splendid biography [4] gives the fascinating details.
Chapter 1

The real line

1.1 Why do we bother?
It is surprising how many people think that analysis consists in the di¬cult
proofs of obvious theorems. All we need know, they say, is what a limit is,
the de¬nition of continuity and the de¬nition of the derivative. All the rest
is ˜intuitively clear™1 .
If pressed they will agree that the de¬nition of continuity and the de¬ni-
tion of the derivative apply as much to the rationals Q as to the real numbers
R. If you disagree, take your favorite de¬nitions and examine them to see
where they require us to use R rather than Q. Let us examine the workings
of our ˜clear intuition™ in a particular case.
What is the integral of t2 ? More precisely, what is the general solution of
the equation

g (t) = t2 ? (*)

We know that t3 /3 is a solution but, if we have been well taught, we know
that this is not the general solution since

t3
g(t) = + c, (**)
3
with c any constant is also a solution. Is (——) the most general solution of
(—)?
If the reader thinks it is the most general solution then she should ask
herself why she thinks it is. Who told her and how did they explain it? If the
1
A good example of this view is given in the book [9]. The author cannot understand the
problems involved in proving results like the intermediate value theorem and has written
his book to share his lack of understanding with a wider audience.


1
2 A COMPANION TO ANALYSIS

reader thinks it is not the most general solution, then can she ¬nd another
solution?
After a little thought she may observe that if g(t) is a solution of (—) and
we set
t3
f (t) = g(t) ’
3
then f (t) = 0 and the statement that (——) is the most general solution of
(—) reduces to the following theorem.
Theorem 1.1.1. (Constant value theorem.) If f : R ’ R is di¬eren-
tiable and f (t) = 0 for all t ∈ R, then f is constant.
If this theorem is ˜intuitively clear™ over R it ought to be intuitively clear
over Q. The same remark applies to another ˜intuitively clear™ theorem.
Theorem 1.1.2. (The intermediate value theorem.) If f : R ’ R is
continuous, b > a and f (a) ≥ 0 ≥ f (b), then there exists a c with b ≥ c ≥ a.
such that f (c) = 0.
However, if we work over Q both putative theorems vanish in a pu¬ of
smoke.
Example 1.1.3. If f : Q ’ Q is given by
if x2 < 2,
f (x) = ’1
f (x) = 1 otherwise,
then
(i) f is a continuous function with f (0) = ’1, f (2) = 1, yet there does
not exist a c with f (c) = 0,
(ii) f is a di¬erentiable function with f (x) = 0 for all x, yet f is not
constant.
Sketch proof. We have not yet formally de¬ned what continuity and di¬eren-
tiability are to mean. However, if the reader believes that f is discontinuous,
she must ¬nd a point x ∈ Q at which f is discontinuous. Similarly, if she
believes that f is not everywhere di¬erentiable with derivative zero, she must
¬nd a point x ∈ Q for which this statement is false. The reader will be in-
vited to give a full proof in Exercise 1.3.5 after continuity has been formally
de¬ned.
The question ˜Is (——) the most general solution of (—)?™ now takes on a
more urgent note. Of course, we work in R and not in Q but we are tempted
to echo Acton ([1], end of Chapter 7).
3
Please send corrections however trivial to twk@dpmms.cam.ac.uk

This example is horrifying indeed. For if we have actually seen
one tiger, is not the jungle immediately ¬lled with tigers, and
who knows where the next one lurks.

Here is another closely related tiger.

Exercise 1.1.4. Continuing with Example 1.1.3, set g(t) = t + f (t) for all
t. Show that g (t) = 1 > 0 for all t but that g(’8/5) > g(’6/5).

Thus, if we work in Q, a function with strictly positive derivative need
not be increasing.
Any proof that there are no tigers in R must start by identifying the dif-
ference between R and Q which makes calculus work on one even though it
fails on the other. Both are ˜ordered ¬elds™, that is, both support operations
of ˜addition™ and ˜multiplication™ together with a relation ˜greater than™ (˜or-

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