# Book:K.G. Binmore/Mathematical Analysis: A Straightforward Approach

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

Published $\text {1977}$, Cambridge University Press

ISBN 0 521 29167 4.

### 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
1.13 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 sequences
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 $\map f x \to l$ as $x \to \xi$
8.6 Continuity at a point
8.8 Connexion with convergent sequences
8.11 Properties of limits
8.16 Limits of composite functions
8.18 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
17.5 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
Notation
Index

Next

## Rating

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

## Errata

### Limit of Root of Positive Real Number

Chapter $\S 5$: Subsequences
$\S 5.4$: Example
In example $4.4$ we showed that, if $x > 0$, then $x^{1 / n} \to 1$ as $n \to \infty$. We prove this again by a different method. As in example $4.4$ we need only consider the case $x \ge 1$.

### Convergent Real Sequence: Exercise: $x_{n + 1} = \dfrac k {1 + x_n}$

Chapter $\S 5$: Subsequences
$\S 5.7$: Solution to Exercise $(3)$
It is easily shown by induction that $k > x_n > 0$ ($n = 1, 2, \ldots$).