Below is an approximate outline for the course. The outline is
"approximate" in the sense that it the course material will follow
the topics listed below, however we may decide to cover some
topics in more depth, or add some more advanced topics at the end
of the course. There is some space in the last couple of
weeks for this.
After class I'll update it with what we covered that day and any
relevant homework assignments, so the "approximate outline" will
become an "exact record" of the lectures.
| Date | Topics covered | Recommended Reading | Homework Assignments |
| Monday January 14 |
Class Introduction Examples of different types of proof Sec 10: Natural numbers and induction |
Read Theorem 10.2 on p100, and work through Examples 10.3 and 10.4 on pp101-102. | |
| Wednesday January 16 |
Sec 10: Natural numbers and induction (cont.) | ||
| Friday January 18 |
More proof by induction Sec 8: Cardinality |
Read Definitions 8.1 and 8.6 on pp78-79. Read Example 8.7 and do Practice Problem 8.8. |
|
| Monday January 21 |
NO CLASS |
||
| Wednesday January 23 |
Sec 8: Cardinality | Read Theorem 8.10 and work through Example 8.11 | |
| Friday January 25 |
Sec 8: Cardinality Sec 11: Ordered Fields |
Read the proof of Theorem 8.12 (Cantor's diagonalisation
argument) and do Practice Problem 8.13. Read the proof of Theorem 8.9. |
Due Friday January 31 Sec 10: 10.1, 10.2, 10.4, 10.5, 10.6, 10.11, 10.14, 10.15, 10.16, 10.17 |
| Monday January 28 |
Sec 11: Ordered Fields (cont.) | Familiarise yourself with the axioms of an ordered field. Work through Practice Problems 11.2 and 11.3. |
|
| Wednesday January 30 |
Problem Session | Work through any remaining problems from the worksheet. | |
| Friday January 31 |
Sec 11: Ordered Fields (cont.) | Due Friday February 8 Sec 8: 8.1, 8.2, 8.3(a)(b)(d), 8.7, 8.9, 8.16, 8.18 |
|
| Monday February 4 |
Sec 12: The Completeness Axiom | ||
| Wednesday February 6 |
Sec 12: The Completeness Axiom (cont.) | ||
| Friday February 8 |
Density of the rationals Sec 13: Topology of the Reals |
Due Friday February 15 Sec 11: 11.1, 11.2, 11.3(c)(f)(h)(j)(k), 11.5, 11.7, 11.10, 11.11, 11.12 Sec 12: 12.1, 12.2, 12.6, 12.7 |
|
| Monday February 11 |
Sec 13: Topology of the Reals (cont.) | ||
| Wednesday February 13 |
Sec 13: Topology of the Reals (cont.) | ||
| Friday February 15 |
Sec 13: Topology of the Reals (cont.) Sec 14: Compact Sets |
||
| Monday February 18 |
Sec 14: Compact Sets (cont.) | ||
| Wednesday February 20 |
Sec 14: Compact Sets (cont.) | ||
| Friday February 22 |
Problem Session | Work through any remaining problems from the worksheet. | Due Friday March 1 Sec 12: 12.10, 12.12, 12.14, 12.15 Sec 13: 13.1, 13.2, 13.3, 13.4, 13.11, 13.15, 13.20 |
| Monday February 25 |
Sec 16: Convergence | You must understand Definition 16.2. This is the most important definition for the entire course! | Recommended problems from Section 14 (for midterm
preparation) 14.1, 14.2, 14.3, 14.4, 14.5(a), 14.6, 14.8(b) |
| Wednesday February 27 |
MIDTERM | ||
| Friday March 1 |
Sec 16: Convergence (cont.) | ||
| Monday March 4 |
Sec 17: Limit Theorems | ||
| Wednesday March 6 |
Sec 17: Limit Theorems (cont.) | ||
| Friday March 8 |
Sec 17: Limit Theorems (cont.) Sec 18: Monotone sequences and Cauchy sequences |
Due Friday March 15 Sec 14: 14.1, 14.2, 14.3, 14.5(a), 14.6 Sec 16: 16.1, 16.2, 16.4, 16.5, 16.6(b)(d)(e), 16.7(a)(f), 16.8(c), 16.9 |
|
| Monday March 11 |
Sec 18: Monotone sequences and Cauchy sequences | ||
| Wednesday March 13 |
Sec 19: Subsequences | ||
| Friday March 15 |
Sec 19: Subsequences (cont.) Sec 20: Limits of functions |
Due Friday March 22 Sec 16: 16.11, 16.15 Sec 17: 17.1, 17.2, 17.3, 17.6, 17.8, 17.15(a) Sec 18: 18.1, 18.2, 18.3(a), 18.4, 18.7, 18.14, 18.15 Sec 19: 19.1, 19.2, 19.5(a)(d)(f), 19.11, 19.12, 19.13 |
|
| Monday March 18 |
Sec 20: Limits of functions | ||
| Wednesday March 20 |
In-class problem session | Click here for the worksheet. It is especially important that you work through the problem on the Cantor set. | |
| Friday March 22 |
Review | ||
| Week of March 25-March 29 |
SPRING BREAK | ||
| Monday April 1 |
Sec 20: Limits of functions (cont.) | ||
| Wednesday April 3 |
Sec 20: Limits of functions (cont.) Sec 21: Continuous functions |
Check out Cauchy's
Wrong Proof. Here is another interesting historical article about the origins of epsilon and delta techniques. |
|
| Friday April 5 |
Sec 21: Continuous functions (cont.) | Work through the proof of Theorem 21.14 | Due Friday April 12 Sec 20: 20.1, 20.2, 20.4, 20.5, 20.7, 20.14, 20.16, 20.18 Sec 21: 21.1, 21.2, 21.3, 21.6(c)(d)(f), 21.11 |
| Monday April 8 |
Sec 21: Continuous functions (cont.) | ||
| Wednesday April 10 |
Sec 22: Properties of continuous functions | Work through examples 22.7 and 22.8 | |
| Friday April 12 |
Sec 22: Properties of continuous
functions (cont.) Sec 23: Uniform continuity |
||
| Monday April 15 |
Sec 25: The Derivative | You must understand Definition 25.1. Work through Examples 25.2 and 25.4, and do Practice Problem 25.5. |
|
| Wednesday April 17 |
Sec 25: The Derivative (cont.) Sec 26: The Mean Value Theorem |
Read Example 26.4 and do Practice Problem 26.5. | |
| Friday April 19 | Sec 26: The Mean Value Theorem (cont.) |
Due Friday April 26 Sec 22: 22.1, 22.2, 22.3(a)(b)(e), 22.5, 22.7, 22.13 Sec 23: 23.1, 23.2, 23.3(d)(e), 23.5 |
|
| Monday April 22 |
Sec 26: The Mean Value Theorem (cont.) Sec 29: The Riemann Integral |
Recommended Problems Sec 25: 25.1, 25.2, 25.5, 25.6, 25.7(d), 25.8, 25.11, 25.15 Sec 26: 26.1, 26.2, 26.5(a)(b)(d)(g)(h)(j), 26.6, 26.10, 26.11, 26.15 |
|
| Wednesday April 24 |
Problem Session | Work through any remaining problems from the worksheet. | |
| Friday April 26 |
Sec 29: The Riemann Integral(cont.) Sec 30: Properties of the Riemann Integral |
||
| Monday April 29 |
Sec 30: Properties of the Riemann Integral (cont.) | ||
| Wednesday May 1 |
Sec 31: The Fundamental Theorem of Calculus | Recommended Problems Sec 29: 29.1, 29.2, 29.5, 29.7, 29.8, 29.9, 29.11, 29.13, 29.16 Sec 30: 30.1, 30.2, 30.3, 30.9, 30.11, 30.12, 30.15, 30.21 Sec 31: 31.1, 31.2, 31.4, 31.5, 31.12, 31.13 |
|
| Friday May 3 |
Review |