Lectures are MW 12:30PM - 1:45PM. Page numbers refer to Rudin, Principles of Mathematical Analysis, 3rd edition.
No class on Labor Day (Mon 9/7), Fall Break (Mon 10/12), and the Thanksgiving recess (Wed 11/25). The last day of classes is Mon 12/14.
There is an extra review session on the evening of Friday 10/23, in addition to the regular lectures. Time and room TBA.
The Homework column shows roughly where each set falls in the term. Due dates are on the homework page.
| Week | Week of | Topic | Reading assignment | Homework |
|---|---|---|---|---|
| 1 | 8/31 | Ordered sets, fields, the real field (pg. 1-8) | The real field, the extended real number system, the complex field, Euclidean spaces (pg. 9-17) | |
| 2 | 9/7 | (no class Mon 9/7) The real field, the extended real number system, the complex field (pg. 9-16) | Finite, countable, and uncountable sets, metric spaces (pg. 24-36) | Homework 1 |
| 3 | 9/14 | Euclidean spaces, finite, countable, and uncountable sets, metric spaces (pg. 16-17, 24-31) | Compact sets (pg. 36-40) | Homework 2 |
| 4 | 9/21 | Metric spaces, compact sets, connected sets (pg. 32-40, 42-43) | Numerical sequences (pg. 47-58) | Homework 3 |
| 5 | 9/28 | Compact sets, perfect sets, the Cantor set, connected sets (pg. 36-43); convergent sequences, subsequences (pg. 47-52) | Cauchy sequences, completeness, upper and lower limits (pg. 52-58) | Homework 4 |
| 6 | 10/5 | Cauchy sequences, complete metric spaces, monotonic sequences, upper and lower limits, some special sequences (pg. 52-58) | Series, the Cauchy criterion, comparison, geometric series (pg. 58-61) | Homework 5 |
| 7 | 10/12 | (no class Mon 10/12) Series, the Cauchy criterion, series of nonnegative terms, comparison, geometric series, Cauchy condensation (pg. 58-61) | The number e, the root and ratio tests (pg. 61-69) | Homework 6 |
| 8 | 10/19 | The p-series, the number e, the root test, the ratio test (pg. 61-69); power series and the radius of convergence, summation by parts, Dirichlet’s test, alternating series (pg. 69-73). Extra review session Fri 10/23. | Absolute convergence, operations with series: addition, multiplication, rearrangements (pg. 71-78) | Homework 7 |
| 9 | 10/26 | Midterm exam (Mon 10/26); absolute convergence, operations with series, products of series, rearrangements (pg. 71-78) | Limits of functions, continuous functions, continuity and compactness (pg. 83-93) | |
| 10 | 11/2 | Limits of functions, continuous functions, continuity and compactness (pg. 83-93) | Continuity and connectedness, discontinuities, monotonic functions, infinite limits and limits at infinity (pg. 93-98) | Homework 8 |
| 11 | 11/9 | Continuity and connectedness, discontinuities, monotonic functions, infinite limits and limits at infinity (pg. 93-98) | The derivative of a real function, mean value theorems, the continuity of derivatives, L’Hospital’s rule (pg. 103-110) | Homework 9; midterm corrections due |
| 12 | 11/16 | The derivative of a real function, mean value theorems, the continuity of derivatives, L’Hospital’s rule, derivatives of higher order, differentiation of vector-valued functions (pg. 103-113) | Definition and existence of the Riemann-Stieltjes integral (pg. 120-129) | Homework 10 |
| 13 | 11/23 | (no class Wed 11/25) Definition and existence of the Riemann-Stieltjes integral (pg. 120-126) | Properties of the integral (pg. 128-134) | |
| 14 | 11/30 | Properties of the integral, integration and differentiation, integration of vector-valued functions, rectifiable curves (pg. 128-137) | The main problem for sequences and series of functions, uniform convergence (pg. 143-151) | Homework 11 |
| 15 | 12/7 | The main problem for sequences and series of functions, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the Weierstrass theorem (pg. 143-153, 159-160) | Review for the final exam | |
| 16 | 12/14 | Review session for the final exam (Mon 12/14, last day of classes) |