ProofWiki:Books/K.G. Binmore/Mathematical Analysis: A Straightforward Approach

K.G. Binmore: Mathematical Analysis: A Straightforward Approach

Published 1977, Cambridge University Press.

ISBN 0 521 29167 4

Subject Matter

• Real Analysis

Contents

Preface

1 Real Numbers

1.1 Set notation

1.2 The set of real numbers

1.3 Arithmetic

1.4 Inequalities

1.9 Roots

1.10 Quadratic equations

1.13 Irrational numbers

1.14 Modulus

2 Continuum Property

2.1 Achilles and the tortoise

2.2 The continuum property

2.6 Supremum and infimum

2.7 Maximum and minimum

2.9 Intervals

2.11 Manipulations with $\sup$ and $\inf$

3 Natural Numbers

3.1 Introduction

3.2 Archimedean property

3.7 Principle of Induction

4 Convergent Series

4.1 The bulldozers and the bee

4.2 Sequences

4.4 Definition of convergence

4.7 Criteria for convergence

4.15 Monotone sequences

4.21 Some simple properties of convergent sequences

4.26 Divergent sequences

5 Subsequences

5.1 Subsequences

5.8 Bolzano-Weierstrass theorem

5.12 Lim sup and lim inf

5.16 Cauchy sequences

6 Series

6.1 Definitions

6.4 Series of Positive Terms

6.7 Elementary properties of series

6.12 Series and Cauchy sequences

6.20 Absolute and conditional convergence

6.23 Manipulations with series

7 Functions

7.1 Notation

7.6 Polynomial and rational functions

7.9 Combining functions

7.11 Inverse functions

7.13 Bounded functions

8 Limits of functions

8.1 Limits from the left

8.2 Limits from the right

8.3 $f \left({x}\right) \to l$ as $x \to \xi$

8.6 Continuity at a point

8.8 Connexion with convergent sequences

8.11 Properties of limits

8.16 Divergence

9 Continuity

9.1 Continuity on an interval

9.7 Continuity property

10 Differentiation

10.1 Derivatives

10.2 Higher derivatives

10.4 More notation

10.5 Properties of differentiable functions

10.12 Composite functions

11 Mean value theorems

11.1 Local maxima and minima

11.3 Stationary points

11.5 Mean value theorem

11.9 Taylor's theorem

12 Monotone functions

12.1 Definitions

12.3 Limits of monotone functions

12.6 Differentiable monotone functions

12.9 Inverse functions

12.11 Roots

12.13 Convex functions

13 Integration

13.1 Area

13.2 The integral

13.3 Some properties of the integral

13.9 Differentiation and integration

13.16 Riemann integral

13.19 More properties of the integral

13.27 Improper integrals

13.31 Euler-Maclaurin summation formula

14 Exponentiation and logarithm

14.1 Logarithm

14.4 Exponential

14.6 Powers

15 Power series

15.1 Interval of convergence

15.4 Taylor series

15.7 Continuity and differentiation

16 Trigonometric functions

16.1 Introduction

16.2 Sine and cosine

16.4 Periodicity

17 The gamma function

17.1 Stirling's formula

17.3 The gamma function

17.8 Properties of the gamma function

18 Appendix

This contains the proofs of 'propositions' left unproved in the main body of the text.

Solutions to exercises

Suggested further reading

Notation

Index

Rating

Speed	2 - Trundle
Clarity	4 - Straightforward
Density	3 - Meaty
Level	2 - Hilly
Scope	3 - Focused
Solutions	5 - Complete