# Course Outline

This is a readable calendar spine transcribed from the Fall 2026 syllabus. Detailed theorem numbering and exercises remain in the Rudin–Ross course handout.

| Week | Topics | Handout classes |
|---:|---|---:|
| 1 | Ordered sets, fields, least-upper-bound property | 1 |
| 2 | The real field; Euclidean spaces; countability | 2–3 |
| 3 | Metric spaces; open and closed sets; compactness | 4–5 |
| 4 | Perfect and connected sets; sequences in metric spaces | 6–7 |
| 5 | Cauchy sequences and completeness; upper and lower limits | 8–9 |
| 6 | Series; comparison, root, and ratio tests; power series | 10–11 |
| 7 | Absolute convergence and rearrangements; limits and continuity | 12–13 |
| 8 | Continuity and compactness; midterm | 14 |
| 9 | Connectedness and monotone functions; derivative and mean value theorems | 15–16 |
| 10 | L’Hospital and Taylor; Riemann–Stieltjes integral | 17–18 |
| 11 | Properties of the integral; fundamental theorem of calculus | 19–20 |
| 12 | Sequences of functions; uniform convergence; `C(X)` | 21–22 |
| 13 | Integration and differentiation of limits; equicontinuity | 23–24 |
| 14 | Arzelà–Ascoli; Weierstrass approximation | 25–26 |
| 15 | Stone–Weierstrass; power series as functions; elementary transcendental functions | 27–29 |
| RRR | Review; optional Fourier-series and gamma-function survey | 30–32 |