### Announcements:

• The final exam is on December 8, 8:30 AM, in SWNG 222.
• Office hours during the exam period: Mondays and Wednesdays, 1-2 pm, in MATH 200. These are for both of my classes, but on Dec. 5 and 7, priority will be given to Math 320 students.
• For practice, here is a final exam from 2015. Solutions are posted here. The last two questions concern differentiation which was not covered in this year's class, so you can skip these.
• Here is a list of the topics we covered. You will be responsible for all of them on the final exam:
• Chapter 1: all of it except for the Appendix, pp. 17-21.
• Chapter 2: everything except "Perfect sets", pp. 41-42
• Chapter 3: we skipped the proofs of Theorems 3.31 and 3.32 (the theorems are useful to know but you don't need to remember the proofs). Theorem 3.37 clarifies the relationship between the root test and the ratio test, but will not be required on the exam. You do not need to remember the summation by parts formula in Theorem 3.41. We skipped the last part of Chapter 3, starting with page 72. ("Absolute convergence" is still required, "Addition and multiplication of series" and "Rearrangements" (pp. 72-78) are not.)
• Chapter 4: we skipped Theorem 4.20. "Infinite limits" (pp. 97-98) was not discussed in class. This is a very easy extension of the concept of limits and you should read it on your own.
• All past homework assignments and solutions are posted here.
• Midterm solutions are here.
• Midterm 1 was on Friday Oct. 21, in class. It covered all material up to and including open and closed sets in metric spaces (ending on p. 35).
• Practice midterm solutions are here.
• A practice midterm is posted here. Solutions will be posted on Wednesday. Update, Oct. 4: In #5(a) and (b), we will expect proofs from the definition. In #5 (c), you may use theorems from the textbook, including the limits in "Some special sequences", pp. 57-58. This material will be covered on Friday, Oct. 7. In #1, we expect you to provide a function that establishes the equality of cardinal numbers, and a proof that this function is a bijection. A picture is not sufficient.
• We are covering Chapters 2 and 3 of Rudin according to the following schedule:
• Chapter 2: "Finite, countable, and uncountable sets", pp. 24-30
• Chapter 3: "Convergent Sequences", pp. 47-51, but for sequences in R instead of general metric spaces.
• Monotone sequences, 3.13-3.14, p. 55
• Subsequences, Def, 3.13, p. 51
• Every bounded sequence in R has a monotone subsequence (not in Rudin), therefore a bounded sequence has a convergent sequence (this corresponds to 3.6 in Rudin, but with a different proof)
• Cauchy sequences and completeness of R, pp. 52-54.
• "Some special sequences", p. 57, needed for homework
• Chapter 2: Metric spaces, pp. 30-43
• Chapter 3: "Upper and lower limits", pp. 55-57
• Chapter 3: "Series" and the rest of the chapter, pp. 58-78. Some topics may be omitted.

## Mathematics 320 (Real Variables I), Fall 2016

Section 102: Prof. I. Laba, MWF 9:00-9:50, BUCH A103
• Instructor contact information: Math Bldg 200, (604) 822 4457, ilaba@math.ubc.ca
• Office hours: Mon 1-2, Wed 10-11, Fri 12-1
• The best way to contact the instructor is by email. Please note that email received on evenings and weekends will be answered on the next business day. If you cannot attend regular office hours due to schedule conflict, please make an appointment in advance. Drop-ins and same-day requests for appointments cannot always be accommodated.
Textbook: W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill,

Calendar description: The real number system; real Euclidean n-space; open, closed, compact, and connected sets; Bolzano-Weierstrass theorem; sequences and series. Continuity and uniform continuity. Differentiability and mean-value theorems.

Prerequisites: Either (a) a score of 68% or higher in MATH 226 or (b) one of MATH 200, MATH 217, MATH 226, MATH 253, MATH 263 and a score of 80% or higher in MATH 220.

Course web pages:
Detailed syllabus (for both sections): available here

Your course mark will be based on homework (30%), one midterm exam (20%), and the final exam (50%). The grades may be slightly scaled at the end of the course.

Examinations: There will be one in-class 50-minute midterm, scheduled on Friday, October 21, and a 2.5 hour final exam in December. The date of the final examination will be announced by the Registar later in the term. Attendance at the final examination is required, so be careful about making other committments (such as travel) before this date is confirmed. All examinations will be strictly closed-book: no formula sheets, calculators, or other aids will be allowed.

Homework assignments: All problem sets and solutions will be posted here. Homework will be assigned weekly or biweekly, depending on the pace of the course. Each homework will be announced and posted here at least a week in advance. The homeworks are due in class on the due date. If you cannot come to class, you may drop off your homework at your instructor's office on the day before it is due. Late assignments will not be accepted. Solutions will be posted on the course webpage immediately after the lecture. To allow for minor illnesses and other emergencies, the lowest homework score will be dropped.

Solutions to textbook problems: There are solutions to Rudin's exercises available on many websites, for example here. You can use these for your own practice, but please be aware that we have not checked any of them and cannot vouch for their correctness. Use your own judgement.

Academic concession: Missing a midterm, or handing in a homework after the deadline, will result in a mark of 0. Exceptions may be granted in two cases: prior consent of the instructor, or a documented medical reason. Your course mark will then be based on your remaining coursework.