Week 0/1: Sections 1.1, 1.2, 1.3, and 1.4 of Abbott and Chapter 1 of Rudin.

Week 2: Sections 3.1, 3.2, and 3.3 of Abbott and Chapter 2 (through the end of the section on perfect sets) in Rudin.

Week 3: Sections 3.4, 3.5, and 3.6 of Abbott and remaining portion of Chapter 2 in Rudin.

Week 4: Sections 2.1-2.6 of Abbott and Pages 47-62 of Rudin.

Week 5: Remaining portion of Chapter 2 in Abbott and Pages 59-79 in Rudin.

Week 6/7: Read Chapter 4 in Abbott and Chapter 4 in Rudin.

Week 8: Read Chapter 5 in Abbott and Chapter 5 and Pages 205-220 in Rudin.

Week 10ish: Read Pages 1-20 in the supplementary course notes.

For those of you curious about the construction of real numbers, the following provides an alternative construction to cuts; I personally think it's more intuitive.

For those of you curious about the history of compactness, take a look at this article: