Mathematical Analysis and Proof

David S G Stirling

• Publisher: Woodhead Publishing
• Year: 2009
• ISBN:
  9780857099341
• Edition: 2
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills are developed with care and then brought to bear on the core material of analysis in such a lucid presentation that the development reads naturally and in a straightforward progression. Retaining the core text, the second edition has additional worked examples which users have indicated a need for, in addition to more emphasis on how analysis can be used to tell the accuracy of the approximations to the quantities of interest which arise in analytical limits.

• Addresses a lack of familiarity with formal proof, a weakness observed among present-day mathematics students
• Examines the idea of mathematical proof, the need for it and the technical and logical skills required

1

Setting the Scene

“If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.”

Francis Bacon.

1.1 Introduction

We have all seen mathematical formulae like

$1+2+3+\dots +\mathit{n}=\frac{1}{2}\mathit{n}\left(\mathit{n}+1\right)$

or

$sin\left(\mathit{x}+\mathit{y}\right)=sin\mathit{x}cos\mathit{y}+cos\mathit{x}sin\mathit{y}.$

What do these actually mean? – and why should we believe them?

In both cases, we mean more than we have said. The first statement has n in it, which we have not explained, but it is understood that n is some positive whole number. If n is a specific positive integer (using the more imposing word integer for a whole number), say 4, then it is easy to check that 1 + 2 + 3 + 4 and $\frac{1}{2}4\left(4+1\right)$

both equal 10. However, we would usually interpret the statement not as being true for one particular value of n but for all positive integers. Obviously, we can test this for as many different values of n as we like but, however many we test, there will remain lots of integers for which the formula has not been tested. Mathematics gives us reasons to believe that the formula is true also for these untested values and that there will be no surprises. The old proverb that “the exception proves the rule” is not part of the mathematical folklore!

The second statement involves two complicated functions, sin and cos, and we shall gloss over the detail of what these mean. The formula here claims that if we choose x and y to be two numbers, which do not have to be integers, then the formula holds for these values. Again there is an argument, more complicated in this case, why we should believe this. This formula, however, holds for a wider range of values of the “variable” x (and a second “variable” y) in that x can ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• Author’s Preface
• 1: Setting the Scene
• 2: Logic and Deduction
• 3: Mathematical Induction
• 4: Sets and Numbers
• 5: Order and Inequalities
• 6: Decimals
• 7: Limits
• 8: Infinite Series
• 9: The Structure of the Real Number System
• 10: Continuity
• 11: Differentiation
• 12: Functions Defined by Power Series
• 13: Integration
• 14: Functions of Several Variables
• Appendix
• Hints and Solutions to Selected Problems
• Notation Index
• Subject Index