LECTURE NOTES OF WILLIAM CHEN
# FUNDAMENTALS OF ANALYSIS

### Chapter 1 : THE NUMBER SYSTEM >>

### Chapter 2 : SEQUENCES AND LIMITS >>

### Chapter 3 : SERIES >>

### Chapter 4 : FUNCTIONS AND CONTINUITY >>

### Chapter 5 : DIFFERENTIATION >>

### Chapter 6 : THE RIEMANN INTEGRAL >>

### Chapter 7 : FURTHER TREATMENT OF LIMITS >>

### Chapter 8 : UNIFORM CONVERGENCE >>

This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in analysis. Chapters 1 - 4 and 7 - 8 were used in various forms and on many occasions between 1982 and 1990 by the author at Imperial College, University of London. Chapters 5 and 6 were added in Sydney in 1994 and 1996 respectively.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- The Real Numbers
- Completeness of the Real Numbers
- The Complex Numbers
- Countability
- Cardinal Numbers

- Introduction
- Real Sequences
- Tests for Convergence
- Recurrence Relations
- Subsequences

- Introduction
- Real Series
- Complex Series
- Power Series
- Multiplication of Series

- Limits of Functions
- Continuity in Intervals
- Continuity in Closed Intervals

- Introduction
- Some Important Results on Derivatives
- Stationary Points and Second Derivatives
- Series Expansion

- Introduction
- Properties of the Riemann Integral
- Sufficient Conditions for Integrability
- Integration as the Inverse of Differentiation
- An Important Example

- Upper and Lower Limits of a Real Sequence
- Double and Repeated Limits
- Infinite Products
- Double Integrals

- Introduction
- Criteria for Uniform Convergence
- Consequences of Uniform Convergence
- Application to Power Series