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

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 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


 * Suggested further reading


 * Notation


 * Index



Limit of Root of Positive Real Number

 * Chapter $\S 5$: Subsequences
 * $\S 5.4$: Example

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

 * Chapter $\S 5$: Subsequences
 * $\S 5.7$: Solution to Exercise $(3)$