Fundamentals of Analysis

by WWL Chen

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 chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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.

Chapter 1: THE NUMBER SYSTEM (last uploaded on 2 December 2002)

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

Chapter 2: SEQUENCES AND LIMITS (last uploaded on 2 December 2002)

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

Chapter 3: SERIES (last uploaded on 2 December 2002)

• Introduction
• Real Series
• Complex Series
• Power Series
• Multiplication of Series

Chapter 4: FUNCTIONS AND CONTINUITY (last uploaded on 2 December 2002)

• Limits of Functions
• Continuity in Intervals
• Continuity in Closed Intervals

Chapter 5: DIFFERENTIATION (last uploaded on 2 December 2002)

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

Chapter 6: THE RIEMANN INTEGRAL (last uploaded on 2 December 2002)

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

Chapter 7: FURTHER TREATMENT OF LIMITS (last uploaded on 2 December 2002)

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

Chapter 8: UNIFORM CONVERGENCE (last uploaded on 2 December 2002)

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