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

iii

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

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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

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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

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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-