A first Analysis course
John O'Connor
CONTENTS
Introduction
Some set theory
Rationals and irrationals
Functions
Infinity and infinities
Axioms for the Real numbers
Convergence in the Reals
Properties of convergent sequences
Monotonic sequences
Subsequences
Cauchy sequences
Continuity of Real functions
Limits of functions
The epsilon-delta definition
Some horrible functions
Metric spaces: definition and examples
Convergence in a metric space
Convergence in infinite dimensional spaces
Properties of uniform convergence
The Weierstrass approximation theorem
The intermediate value theorem
The boundedness theorems
Images of intervals
Supplements
Definitions of the concept of a function
Some Early History of Set Theory
Dedekind cuts
Farey sequences
Cardinal numbers
Continued fractions
The golden ratio
Space filling curves
More about the delta-function
Exercises
Exercises 1
Exercises 2
Exercises 3
Exercises 4
Exercises 5
Exercises 6
Exercises 7
Exercises 8
Exercises 9
Exercises 10
JOC MT2002 Analysis September 2001
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