Preface
1 Preliminaries
1.1 The Logic cf Quantifiers
1.1.1 Rules of Quantifiers
1.1.2 Examples
1.1.3 Exercises
1.2 Infinite Sets
1.2.1 Countable Sets
1.2.2 Uncountable Sets
1.2.3 Exexcises
1.3 Proofs
1.3.1 How to Discover Proofs
1.3.2 How to Understand Proofs
1.4 The Rational Number System
1.5 The Axiom of Choice*
2 Construction of the Real Number System
2.1 Cauchy Sequences
2.1.1 Motivation
2.1.2 The Definition
2.1.3 Exercises
2.2 The Reals as an Ordered Field
2.2.1 Defining Arithmetic
2.2.2 The Field Axioms
2.2.3 Order
2.2.4 Exercises
2.3 Limits and Completeness
2.3.1 Proof of Completeness
2.3.2 Square Roots
2.3.3 Exercises
2.4 Other Versions and Visions
2.4.1 Infinite Decimal Expansions
2.4.2 Dedekind Cuts*
2.4.3 Non-Standard Analysis*
2.4.4 Constructive Analysis*
2.4.5 Exercises
2.5 Summary
3 Topology of the Real Line
3.1 The Theory of Limits
3.1.1 Limits, Sups, and Infs
3.1.2 Limit Points
3.1.3 Exercises
3.2 Open Sets and Closed Sets
3.2.1 Open Sets
3.2.2 Closed Sets
3.2.3 Exercises
3.3 Compact Sets
3.3.1 Exercises
3.4 Summary
4 Continuous Functions
4.1 Concepts of Continuity
