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Mathematical Analysis and Proof. Edition No. 2
Description:
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
Contents:
Author's Preface
1: Setting the Scene
1.1 Introduction
1.2 The Common Number Systems
2: Logic and Deduction
2.1 Introduction
2.2 Implication
2.3 Is This All Necessary
or Worthwhile?
2.4 Using the Right Words
3: Mathematical Induction
3.1 Introduction
3.2 Arithmetic Progressions
3.3 The Principle of Mathematical Induction
3.4 Why All the Fuss About Induction?
3.5 Examples of Induction
3.6 The Binomial Theorem
4: Sets and Numbers
4.1 Sets
4.2 Standard Sets
4.3 Proof by Contradiction
4.4 Sets Again
4.5 Where We Have Got To
and The Way Ahead
4.6 A Digression
5: Order and Inequalities
5.1 Basic Properties
5.2 Consequences of the Basic Properties
5.3 Bernoulli's Inequality
5.4 The Modulus (or Absolute Value)
6: Decimals
6.1 Decimal Notation
6.2 Decimals of Real Numbers
5.3 Some Interesting Consequences
7: Limits
7.1 The Idea of a Limit
7.2 Manipulating Limits
7.3 Developments
8: Infinite Series
8.1 Introduction
8.2 Convergence Tests
8.3 Power Series
8.4 Decimals again
Problems
9: The Structure of the Real Number System
Problems
10: Continuity
10.1 Introduction
10.2 The Limit of a Function of a Real Variable
10.3 Continuity
10.4 Inverse Functions
10.5 Some Discontinuous Functions
11: Differentiation
11.1 Basic Results
11.2 The Mean Value Theorem and its Friends
11.3 Approximating the Value of a Limit
12: Functions Defined by Power Series
12.1 Introduction
12.2 Functions Defined by Power Series
12.3 Some Standard Functions of Mathematics
12.4 Further Examples
Problems
13: Integration
13.1 The Integral
13.2 Approximating the Value of an Integral
13.3 Improper Integrals
Problems
14: Functions of Several Variables
14.1 Continuity
14.2 Differentiation
14.3 Results Involving Interchange of Limits
14.4 Solving Differential Equations
Appendix
The Expression of an Integer as a Decimal
Hints and Solutions to Selected Problems
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Notation Index
Subject Index
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