A first Analysis course

John O'Connor


CONTENTS

Introduction
  1. Some set theory
  2. Rationals and irrationals
  3. Functions
  4. Infinity and infinities
  5. Axioms for the Real numbers
  6. Convergence in the Reals
  7. Properties of convergent sequences
  8. Monotonic sequences
  9. Subsequences
  10. Cauchy sequences
  11. Continuity of Real functions
  1. Limits of functions
  2. The epsilon-delta definition
  3. Some horrible functions
  4. Metric spaces: definition and examples
  5. Convergence in a metric space
  6. Convergence in infinite dimensional spaces
  7. Properties of uniform convergence
  8. The Weierstrass approximation theorem
  9. The intermediate value theorem
  10. The boundedness theorems
  11. Images of intervals

Supplements
  1. Definitions of the concept of a function
  2. Some Early History of Set Theory
  3. Dedekind cuts
  4. Farey sequences
  5. Cardinal numbers
  6. Continued fractions
  7. The golden ratio
  8. Space filling curves
  9. More about the delta-function
Exercises
  1. Exercises 1
  2. Exercises 2
  3. Exercises 3
  4. Exercises 4
  5. Exercises 5
  6. Exercises 6
  7. Exercises 7
  8. Exercises 8
  9. Exercises 9
  10. Exercises 10
JOC MT2002 Analysis September 2001