Chapter 1 The Real Numbers
- 1.1 Discussion: The Irrationality of $\sqrt 2$ (no exercises)
- 1.2 Some Preliminaries
- 1.3 The Axiom of Completeness
- 1.4 Consequences of Completeness
- 1.5 Cardinality
- 1.6 Cantor’s Theorem
- 1.7 Epilogue (no exercises)
Chapter 2 Sequences and Series
- 2.1 Discussion: Rearrangements of Infinite Series (no exercises)
- 2.2 The Limit of Sequence
- 2.3 The Algebraic and Order Limit Theorem
- 2.4 The Monotone Convergence Theorem and a First Look at Infinite series
- 2.5 Subsequences and the Bolzano-Weierstrass Theorem
- 2.6 The Cauchy Criterion
- 2.7 Properties of Infinite Series
- 2.8 Double Summations and Products of Infinite Series
- 2.9 Epilogue (no exercises)
Chapter 3 Basic Topology of $\mathbf R$
- 3.1 Discussion: The Cantor Set (no exercises)
- 3.2 Open and Closed Sets
- 3.3 Compact Sets
- 3.4 Perfect Sets and Connected Sets
- 3.5 Baire’s Theorem
- 3.6 Epilogue (no exercises)
Chapter 4 Functional Limits and Continuity
- 4.1 Discussion: Examples of Dirichlet and Thomae (no exercises)
- 4.2 Functional Limits
- 4.3 Continuous Functions
- 4.4 Continuous Functions on Compact Sets
- 4.5 The Intermediate Value Theorem
- 4.6 Sets of Discontinuity
- 4.7 Epilogue (no exercises)
