A Companion to Analysis

A Companion to Analysis

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.'

In order to concentrate

Although I hope this book may be useful to others, I wrote it for students 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 difficulty, show why they are points of difficulty 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.

Calculus

I learned calculus from the excellent Calculus [13] of D. R. Dickinson and its inspiring author. My first glimpse of analysis was in Hardy's Pure Mathematics [24] read when I was too young to really understand it. I learned elementary analysis from Ferrar's A Textbook of Convergence [18] (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].

Cauchy's defiant preface

Everybody can sympathise with Cauchy's students who just wanted to pass their exams and with his colleagues who just wanted the standard material 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 splendid triumphs of the human intellect. We echo Cauchy's defiant preface to his printed lecture notes.

Analysis consists in the difficult proofs

It is surprising how many people think that analysis consists in the difficult proofs of obvious theorems. All we need know, they say, is what a limit is, the definition of continuity and the definition of the derivative. All the rest is 'intuitively clear

Theorem 1.

Theorem 1.1. (Constant value theorem.) /// : Ш. —> M. is differentiable and f'(t) = 0 for all teK, then f is constant.
If this theorem is 'intuitively clear' over Ш. it ought to be intuitively clear over Q. The same remark applies to another 'intuitively clear' theorem.

Sketch proof

We have not yet formally defined what continuity and differentiability are to mean. However, if the reader believes that / is discontinuous, she must find a point x 6 Q at which / is discontinuous.

Exercise 1.4.

Continuing with Example 1.3, set g(t) = t + f(t) for all t. Show that g'(t) = 1 > 0 for all t but that #(-8/5) > #(-6/5).
Thus, if we work in Q, a function with strictly positive derivative need not be increasing.

We choose

Many ways have been tried to make calculus rigorous and several have been successful. We choose the first and most widely used path via the notion of a limit. In theory, my account of this notion is complete in itself. However, my treatment is unsuitable for beginners and I expect my readers to have substantial experience with the use and manipulation of limits.|

Definition 1.5.

We work in an ordered field F. We say that a sequence a\, a,2, • • • tends to a limit a as n tends to infinity, or more briefly
an —»• a as n —»• oo
if, given any e > 0, we can find an integer no (б) [read 'щ depending on e'] such that
\an — a\ < e for all n > no (б).
The following properties of the limit are probably familiar to the reader.

Proof

I shall give the proofs in detail but the reader is warned that similar proofs will be left to her in the remainder of the book.
(i) By definition:-