Elementary Real Analysis
Math 444 Section F13 Spring 2010_______________________________________
Instructor: Prof. Zhong-Jin Ruan (ruan@math.uiuc.edu)
Classroom: 447 Altgeld Hall; MWF 2:00-2:50pm
Office Hour: MW 1-1:50pm, or by appointment.
Office: 353 Altgeld Hall
Web page: http://www.math.uiuc.edu/~ruan/444F13.html
Textbook: Introduction to Real Analysis by R.B.Bartle and D.R.Sherbert, 3rd Edition, John Wiley & Sons, 2000.
Homework: Every Wednesday (except exam weeks) a homework assignment will be due in class.
No late homework will be accepted for any reason. If you have a reasonable excuse for missing an
assignment, I will score it by the average of the other assignments.
Grading policy:There will be total of 500 points computed as follows.
| Homework | 10 x 10 pts | 100 pts |
| Exams | 2 x 100 pts | 200 pts |
| Final Exam | 200 pts | |
| Total | 500 pts |
Your final grade will be based on the total scores.
HOMEWORK ASSIGNMENTS
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HW #1.
Page 11: #5, #13; Page 16: #6, #16, #20
(Due Wednesday, Jan 27)
HW #2.
Page 38: #4, #10; Page 43: #5, #18; Page 50: #7, and the following problem:
Let S = {r : rationals in [0, sqrt 2 )}. Find sup S and justify your answer.
(Due Wednesday, February 3).
HW #3.
Page 59: #5c, #5d, #10 and show that lim(n+1)/(3n-1) = 1/3;
Page 67: #7, #9, #13.
(Due Wednesday, February 10).
HW#4. Page 74, #2 (assume that 1 < x_1 < 2), #4;
Page 80 #4 (Hint: find upper limit and lower limit), #12, #14.
(Due Wednesday, February 17).
1st Exam on Wednesday, February 24
HW#5. Page 84, #3b, #9, #10, and the following problems
1. Let (a_n) and (b_n) be bounded sequences, show limsup (a_n+b_n) <= limsup a_n + limsup b_n.
2. Extra Credit: If lim a_n = a exists, show that limsup (a_n + b_n) = lim a_n + limsup b_n (Due Wednesday, March 3).
HW#6 Page 104, #7 (consider c>0 case), #9a, #10b, #11c, Page 110, #4.(Due Wednesday, March 10).
HW#7 Page 124, #7, #12; Page 129, #7, #8; Page 135, #1, #3, #11.
(Due Wednesday, March 17).
HW#8 Page 144, #4, #7 (show xsin(x) is not uniformly continuous on R), #8, #9, and #10 (assume that A=(a, b) is a finite open interval).
(Due Wednesday, March 31).
HW#9 Page 175, #6, #11, #13, #14, and the following problem:
Let f be a continuous function defined by f(x) = x^2 sin(1/x)
if x is not 0, and f(0) = 0. Show that f is uniformly
continuous on R.
(Due Wednesday, April 7).
2nd Exam on Monday, April 12
HW#10 Page 202, #13, Page 208, #2, #8, #10, and #12
(Due Wednesday, April 21).
HW#11 Page 217, #3, #6, #10, #11, and the following problem:
Let f be a bounded function on [a, b]. Show that if f is Riemann integrable,
then so is |f|.
(Due Wednesday, April 28).
Final Review: Tuesday, May 11, 4:30-5:30pm at 447AH.
Final Exam: Thursday, May 13, 8-11am at 447 AH.