Mathematical Analysis and Proof

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