| Topic Number | Major Topic | Text Section | Subtopics |
| 1 |
Mathematical Logic |
1.1 (Esty) | Preview of Proof |
| 1.3 (Esty) | Logic for Mathematics |
| 1.4 (Esty) | Important Logical Equivalences |
| 1.5 (Esty) | Negation |
| 1.6 (Esty) | Tautologies and Proofs |
| 2 |
Set Theory, Functions, and Relations |
1.2 (Esty) | Sets |
| (Handout) | Relations |
| (Handout) | Functions |
| 3 |
The Real Numbers |
1.1 | Irrationality of the Square Root of 2 |
| 1.2 | Some Preliminaries |
| 1.3 | The Axiom of Completeness |
| 1.4 | Consequences of Completeness |
| 1.5 | Cantor's Theorem |
| 4 |
Sequences and Series |
2.1 | Rearrangements of Infinite Series |
| 2.2 | The Limit of a Sequence |
| 2.3 | Algebraic and Order Limit Theorems |
| 2.4 | The Monotone Convergence Theorem |
| 2.5 | The Bolzano-Weierstrass Theorem |
| 2.5 | The Cauchy Criterion |
| 2.5 | Properties of Infinite Series |
| 2.5 | Double Summation and Products of Series |
| 5 |
The Basic Topology of R |
3.1 | The Cantor Set |
| 3.2 | Open and Closed Sets |
| 3.3 | Compact Sets |
| 3.4 | Perfect Sets and Connected Sets |
| 3.5 | Baire's Theorem |
| 6 |
Functional Limits and Continuity |
4.1 | The Examples of Dirichlet and Thomae |
| 4.2 | Functional Limits |
| 4.3 | Combinations of Continuous Functions |
| 4.4 | Continuous Functions on Compact Sets |
| 4.5 | The Intermediate Value Theorem |
| 4.6 | Sets of Discontinuity |