Math 444, Section E12, Spring 2008

Math 444, Section E13, Spring 2008

. Lectures: MWF 1:00-1:50 in 143 Henry Bld

. Instructor: Tao Mei, mei@math.uiuc.edu

. Course homepage: www.math.uiuc.edu/~mei/math444.html

. Office & Telephone: ALTGELD Hall 109, 217-244-2478

. Office Hours: 5:00-6:00pm Tuesday or by appointment.

. Textbook (required): R. B. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Edition, John Wiley & Sons, 2000.?

Course Description: This course is an introduction to e - d analysis on the real line for students who do not plan graduate study in mathematics. Students should leave the course not only with a basic understanding of the fundamental concepts of real analysis, but also an improved ability at reading and writing mathematical arguments. Math347 is a prerequisite. Regular homework is an important aspect of the course. See the Approximate Schedule for information of which section will be covered.

Tentative exam dates: There will be two hour-long exams during the semester and a final exam. Books, notes, calculators are not allowed on exams.

Exam #1 7:00-8:00pm, Wednesday, Feb. 27^th in 243 Altgeld Hall

Suggested practicing problems for Exam 1: Quizzes1-2 and Hws 1-5 are enough.

Exam #2 - 7:00-8:00pm, Wednesday, Apr. 9^th in 243 Altgeld Hall

Final Exam- 1:30-4:30 PM, Wednesday, May 7 in our regular classroom (143 Henry building). Please arrange any travel, etc., so that you can take the final on this date. As a "non-combined" course, there will be no conflict exam given except for those few individuals who meet the official university criteria given here in the student code.

Homework and Quiz: Practice is essential for this course. We will have a homework assignment each week. Written homework is due at the beginning of class on the due date. We will also have practicing quizzes. 3 poorest grades of HWs and quizzes will be dropped from the final score in the class. As a result HW extension or make-up quizzes will not be allowed.

Homework Rules: (1)Print your name, section at the top of the paper. (2) Staple the papers together if your homework takes more than one page. (3) Work out the problems, the answer only will not be accepted.(5)Be neat and legible.

You will be able to check your homework and exam grades at Score Reports, which is the Math Department's gradebook program. This will be available beginning approximately two weeks into the semester. All your h/work, quiz and exam grades will be posted there. You will be prompted for your NetID and NetID password after following the Score Reports link. Please check score reports regularly to make sure your grades have been correctly reported and tell me promptly about any errors. You are responsible for keeping all of your graded homework and exams so that any discrepancies in recorded grades can be settled.

Grading: two midterm (20% each), final exam (30%), Homework+Quizzes (30%)

Approximate Course Schedule

Chapter		Class Hours
1.	Preliminaries (We will treat Sections 1.1, 1.2 lightly. Cover Section 1.3.)	3
2.	The Real Numbers (Sections 2.1, 2.2 will be covered quickly, with emphasis placed on 2.3 and 2.4. We may omit the discussion of decimals in Sec. 2.5.)	5
3.	Sequences (We may omit Section 3.6.)	9
4.	Limits (We will omit Section 4.3)	3
5.	Continuous Functions (We will Omit Sections 5.5, 5.6 and omit approximation in (p. 140-144) Section 5.4)	6
6.	Differentiation (We will omit Sections 6.3, 6.4.)	3
7.	The Riemann Integral (We will omit Section 7.4)	6
8.	Sequences of Functions (We sill cover 8.1 and as much of 8.2 as time permits.)

Homework Assignments (An complete assignment ends with a .. An assignment end with a ; is incomplete, additional problems will be added to it later. Each team need to hand in only one homework)

Course hotline (email to students and instructor): Share your questions on homework, course material, homework problems will only be discussed after explicit request.

. Assignment #1 (due by the beginning of the class on Wednesday Jan. 23):

Ex1.1 of textbook: #15, 20,

Ex1.2: #15, Why the poof of claim on page15 is wrong? , Read 1.2.5. on page 15;

Ex1.3: #4, Prove the set Z of all integers is countable, #9, prove Theorem 1.3.9 on page 19 of textbook, Read 1.3.13 on page 21, Construct a surjection from the set of real numbers R to the set P(N) of all subsets of natural numbers (hint: we may consider the function f from R to P(N) with f(x) being the set of all n such that 1 appears in x at the n-th position after the decimal point. For example, for real number x=2.137516899241, we set f(x)={1,5,12}. If 1 does not appear in x after the decimal point, we let f(x)=empty set), Use Theorem 1.3.13 to prove (i) P(N) is uncountable (ii) R is uncountable. (easy fact: every bijection is a surjection, the composition of a surjection and a bijection is still a surjection).

. Assignment #2 (due by the beginning of the class on Wednesday Jan. 30):

(i) Prove that square root of 3 is an irrational number.

(ii) Given a<b in R, find a bijection from (a,b)={a<x<b} to R.

(iii) Ex2.2 #14, 15;

(iv) Ex2.3 #9, Ex2.4 #1.

. Assignment #3 (due by the beginning of the class on Friday Feb. 8th):

(0) Reading assignment: Theorem 2.4.7 on Page 41, Example 2.4.2 on Page 39, the rest of the proof of 2.5.1 on Page 45.

(i) Ex2.4, #4, #18;

(ii) Ex 2.5, #6, #7, #8;

(iii) Ex 3.1 #4, #5(b), #9 (without using 3.2.10).

. Assignment #4 (due by the beginning of the class on Wednesday Feb. 13th):

(0) Reading assignment: Example 3.2.8 (b)(g)(h)

(i) Exercise 3.2 #1bc, #5a, #6, #15ac,#17;

(ii) Exercise 3.3 #1,#4.

. Assignment #5 (due by the beginning of the class on Wednesday Feb. 20th):

(0)Reading assignment: Proof of Theorem 3.5.5, Definition 3.5.7, Proof of Corollary 3.5.9;

(1) Exercise 3.4 #1,#4,#12,#16;

(2)Exercise 3.5 #2, #4,#12.

. Assignment #6 (due by Monday Mar. 3rd):

(0)Reading assignment:

(1)Exercise 3.6 #4a (prove it by definition),#10;

(2) Exercise 3.7 #3a, #4;

(3) Exercise 4.1 #1ac,7 (hint: x^3-c^3=(x-c)(x^2+xc+c)).

. Assignment #7 (due by Wednesday Mar. 12th):

(0)Reading assignment:

(1)Exercise 4.1 # 9ab, #11ac;

(2) Exercise 4.2 #2 (you can use #14 without proving it), #10;

(3) Exercise 5.1#7, #10.

Assignment #8 (due by Wednesday Mar. 26th)

(1) Exercise 5.2 #1,5,6;

(2) Exercise 5.3 #1,5, 6 (hint: For #5, find x_1<x_2<x_3 such that p(x_1)>0,p(x_2)<0,p(x_3)>0 then use our theorem; use the nested interval method introduced in class to locate the roots. For #6, use our theorem for the function g with a=0, b=1/2; think the earth’s equator as a circle then each point on it correspondence to a \theta with 0\leq \theta<2pi. In this situation, 0 and 2pi present the same point. Set f(x) to be the temperature at the point corresponding to 2pi x for 0\leq x\leq 1)

Assignment #9 (due by Wednesday April. 2nd)

(1) Prove that f(x)=square root of x is uniformly continuous on (0,1) but f is not a Lipschitz function.

(2) Exercise 5.4 #2,#3;

(3) Exercise 6.1 #1ab,#3 and prove Theorem 6.1.3(d), #5ab (use quotient rule for (a); for (b), use chain rule for \sqrt x and 5-2x+x^2).

Assignment #10 (due by Wednesday April 16th)

(1) Ex6.1 #15;

(2) Ex6.2 #1ac, #6, #11,#14;

(3) Ex7.1 #2ad, #6.

Assignment #11 (due by Wednesday April 23rd)

(1) Ex7.1 #8,13;

(2) Ex7.2 #2; #7,#10,#12.

Assignment #12 (due by Wednesday April 30)

Ex 7.3 #4,#9ac,#16ac, #20, Prove that f+g+h is Rieman integrable if all f g, h are Rieman integrable by Lebsgue Integration Creterion (hint: use #20) (Assignment completed).