Readings are assigned in Thomson, Brian S., Judith B. Bruckner, and Andrew M. Bruckner. 2008. Elementary Real Analysis. 2nd ed. ISBN: 9781434843678
A screen-optimized PDF version of the textbook is available.
A print-optimized PDF version of the textbook is available.
| Lec # | Lecture Title | Readings |
|---|---|---|
| 1 | Introduction to Real Numbers | Sec. 1.1–1.4 |
| 2 | Introduction to Real Numbers (cont.) | Sec. 1.5–1.6 |
| 3 | How to Write a Proof; Archimedean Property | Sec. 1.7 |
| 4 | Sequences; Convergence | Sec. 2.1–2.2, 2.4–2.7 |
| 5 | Monotone Convergence Theorem | Sec. 2.8, 2.9, 2.11 |
| 6 | Cauchy Convergence Theorem | Sec. 2.12 |
| 7 | Bolzano–Weierstrass Theorem; Cauchy Sequences; Series | Sec. 2.11–2.12, 3.4 |
| 8 | Convergence Tests for Series; Power Series | Sec. 3.5–3.6 |
| 9 | Limsup and Liminf; Power Series; Continuous Functions; Exponential Function | Sec. 2.13, 10.2, 5.4, 5.5 |
| 10 | Continuous Functions; Exponential Function (cont.) | Sec. 5.5, 5.9 |
| 11 | Extreme and Intermediate Value Theorem; Metric Spaces | Sec. 5.4, 13.1–13.2 |
| 12 | Convergence in Metric Spaces; Operations on Sets | Sec. 13.4–13.5 |
| 13 | Open and Closed Sets; Coverings; Compactness | Sec. 13.5 |
| 14 | Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space | <none> |
| 15 | Derivatives; Laws for Differentiation | Sec. 7.2, 7.3 |
| 16 | Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion | Sec. 7.6, 7.11–7.12 |
| 17 | Taylor Polynomials; Remainder Term; Riemann Integrals | Sec. 8.6 |
| 18 | Integrable Functions | Sec. 8.3 |
| 19 | Fundamental Theorem of Calculus | Sec. 8.3 |
| 20 | Pointwise Convergence; Uniform Convergence | Sec. 9.2–9.4 |
| 21 | Integrals and Derivatives under Uniform Convergence | Sec. 9.5–9.6 |
| 22 | Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs) | Sec. 9.5–9.6 |
| 23 | Existence and Uniqueness for ODEs: Picard–Lindelöf Theorem | Sec. 13.11.4 |
