Mathematical Analysis and Proof

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

University students

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