MAT 337 - REAL ANALYSIS

Announcements: Please check here regularly for new announcements.

Jan. 12. ps1, the first problem set is posted below, due Jan. 25. Please remember that the assigned problems are from the current edition of the book, published by Springer, (blue and yellow book) not the old edition that some of you have.

Jan. 14. For those who do not have the current version of the text, there is a second version of ps1 now posted below which contains all the problems referenced in ps1.

Jan. 19. Due to my absence the due date for PS1 is postponed to Wed. Jan. 27. I will hold an extra office hour on Friday Jan. 22, 2-3.

Jan. 27 Office hours have been changed to Mon. and Wed. 1:10 to 2:00 in MP 134, our classroom. These will function also as problem solving sessions, if there are no questions. Please feel free to stay for the office hour and leave whenever you like. I will leave when there is no-one left in the room so do not leave planning to come back lster. For those who can't be there at those times I will also be available in my office Mon and Wed. 2:10 - 2:30.

Jan. 27. ps2 is now posted below and due on Wed. Feb. 9.

Feb. 2. Please see the correction posted below to Problem #2.

Feb.2. Hint for 2.8G on PS2: Say we are trying to show that sup S exists. Start with any a_1 in S and b_1 > S (recall this means b_1 is an upper bound for S ) and let I_1 = [a_1,b_1] . Let c be the midpoint of I_1 . If c > S let I_2 be the (closed) first half of I_1 , otherwise let I_2 be the second half. Repeat the process to get a nested sequence I_n = [a_n,b_n] , show that (a_n) and (b_n) are Cauchy sequences with a common limit L and show that L = sup S.

Feb. 3. Please note the obvious typo in #4 of PS2: |x_n-a| should be d(x_n,a).

Feb. 8. The next assignment is due on Wed. Feb. 10, not Feb. 9.

Feb. 8. Outline for a solution for 9.1 N., part (b), "if" direction. (x_n means x subscript n, and x^a means x to the power a.) Assume the stated condition on balls and suppose that (x_n) is a Cauchy sequence. We must show that (x_n) converges. To this end choose n_1 such d(x_n,x_(n_1)) < 1/2 for all n > n_1 and let E_1 be the closed ball of radius 1 centered at x_(n_1). Then choose n_2 > n_1 such that d(x_n,x_(n_2)) < 1/4 and let E_2 be the closed ball of radius 1/2 centered at x_(n_2). Esplain why E_2 \subset E_1. Continue in this way to find n_1 < n_2 < n_3 < ... so that, if we let E_k be the ball of radius 2^(k-1) centered at x_k, then (E_k) is a decreasing sequence. By our assumption on sequences of closed balls there is at least one point, say a, in the intersection of all the E_k. Show that x_(n_k) converges to a.

Feb. 15. PS3 is now posted (due Feb. 24), as are solutions to PS2.

Feb. 21. Please note the obvious typo in #6 of PS3: in the definition of interior the ball centered at x should be contained in E, not X.

Feb. 22. Last year's Test 1 is posted below. Please remember that we had not covered metric spaces at this point last year!

Feb. 22. Another obvious typo in #3 of PS3: B should be Y.

Mar. 3. Solutions to Test1 are posted below.

Mar. 7. PS4 is posted below, due Mar. 19.

Mar. 22. PS5 is posted below, due Mar. 31. Solutions to PS4 are also posted. The test this Friday will cover all the material in Problem Sets 1 to 4. the sections in DD are Chapters 1-6, complete, 7.1 and 7.2 and Chapter 9, complete.

Mar. 26. The grader will not have time to grade PS5, so it is no longer to be handed in. Please note an error in the solutions to PS4: in the last series problem the limit of a_n/a_(n+1) is less than one so the limit of a_(n+1)/a_n is greater than one and hence the conclusion should be that the series diverges.

Apr. 1. There are review sessions scheduled for Mon. Apr. 5, 10:10 to 12:00, and Mon. Apr.12,12:10 to 2:00. Both will be in BA6183. The second one will be led by Aaron Tikuisis, who has been the grader for the course and took two lectures in January. I will also be available in my office Wed. Apr. 14, 10:10 to 12:00 for short last minute questions.

Apr. 3. I will also be available in my office Wed Apr. 7, 10:00 to 11:00. You can pick up PS4 at that time. I will leave them in a box outside my office door. Any which have not been picked up by late Thursday afternoon I will return to Aaron Tikuisis so you can pick them up at his review session, BA6183 Mon Apr. 12, 12:10 to 2:00. You may also pick them up during my office hour Wed. Apr. 14.

Apr. 5. I have to change my office hours for Wed. Apr. 7 to 9:30 to 10:30.

Apr. 5. The exam will cover all of the course material: Problem Sets 1 through 6. The relevant sections of the text are Chapters 1 through 9, with the exception of 5.7, 7.4 to 7.7 and 8.3.4 (Leibnitz's rule). There are 10 questions on the exam. Two have only one part, four have two parts and four have three parts. the first question has three part but is routine and should take you no more than five minutes. This is all the information I will be giving out about the exam so please don't ask for any more.

Text: Real analysis and applications: theory in practice, by Davidson and Donsig

Sections to be covered: Chapt. 1, 2, 3, 4.1, 9 (all), 4.2-4.4, 5, 7, 8, 10.3-10.6, 11.1, 11.2, 11.7.

Other References

Principles of mathematical analysis, W. Rudin

Real analysis, F. Morgan

Real analysis, N. L. Carothers

Real analysis, Bartle and Sherbert

Real analysis, Haaser and Sullivan (Dover paperback - cheap!)

Elementary real analysis, Bruckner and Thomson

Grading scheme: Term work will consist of 6 problem sets, to be handed in for grading (not all problems will be graded), for a total of 20% of your course grade and two in-class midterms each worth 15% of your course grade. The mid-terms will be held on Fri. Feb. 26 and Fri. Mar. 26, in class. The final exam will be worth 50% of your course grade. If your grade on the final is higher than your calculated course grade then your final grade becomes your course grade.

Problem Sets