Week 3: Read the following in Introduction to Mathematics: Number, Space, and Structure by Scott Taylor: Section 2.4, Chapter 6 (You can read this in any level of detail you want, e.g., skim if you want. Particular attention should be paid to Section 6.3.), and Pages 237-245, 252-255, 264-269 of Chapter 9. In Understanding Analysis by Abbott, read Sections 1.2 and 1.3.
Week 4: Read Ordered Fields from Principles of Mathematical Analysis by W. Rudin. In Understanding Analysis by Abbott, read Section 1.4, 2.1, and 2.2. Also, in Abbott's book, skim Sections 1.5 and 1.6.
Week 5: In Understanding Analysis by Abbott, read Section 2.2, 2.3, and 2.4.
Week 6+7: In Understanding Analysis by Abbott, read Section 2.5, 2.6, and 2.7. Here is a Matlab script which approximates square roots.
Week 8: In Understanding Analysis by Abbott, read Section 3.1, 3.2, and 3.3.
Week 9: In Understanding Analysis by Abbott, read Section 3.4, 4.1, and 4.2