ProofWiki:Books/C.R.J. Clapham/Introduction to Mathematical Analysis

C.R.J. Clapham: Introduction to Mathematical Analysis

Published 1973, Routledge & Kegan Paul.

ISBN 0 7100 7529 4

Subject Matter

• Real Analysis

Contents

Preface

1. Axioms for the Real Numbers

1 Introduction

2 Fields

3 Order

4 Completeness

5 Upper bound

6 The Archimedean property

Exercises

2. Sequences

7 Limit of a sequence

8 Sequences without limits

9 Monotone sequences

Exercises

3. Series

10 Infinite series

11 Convergence

12 Tests

13 Absolute convergence

14 Power series

Exercises

4. Continuous Functions

15 Limit of a function

16 Continuity

17 The intermediate value property

18 Bounds of a continuous function

Exercises

5. Differentiable Functions

19 Derivatives

20 Rolle's theorem

21 The mean value theorem

Exercises

6. The Riemann Integral

22 Introduction

23 Upper and lower sums

24 Riemann-integral functions

25 Examples

26 A necessary and sufficient condition

27 Monotone functions

28 Uniform continuity

29 Integrability of continuous functions

30 Properties of the Riemann integral

31 The mean value theorem

32 Integration and differentiation

Exercises

Answers to the Exercises

Index