Math 311 - 01, Elementary Analysis, Fall 2001

Instructor: Vladimir Bolotnikov

Office: Jones 110, Tel:221-2009, Fax: 221-2988

E-mail: vladi@math.wm.edu, http://www/math.wm.edu/~vladi

Class meetings: T R, 9:30 - 10:50 p.m. in Morton 343

Ofice hours: T R 3:30 - 5:30 p.m. or by appointments

Course Description

    Textbook: Walter Rudin, "Principles of Mathematical Analysis", Third edition. 


    Material to be covered: Parts of Chapters 1-7.

    Homework: It is essential to do mathematics in order to learn it. Homeworks will
              be assigned every lecture. I am ready to answer questions on your 
              homework -  mostly during my office hours (or meetings by appointment). 
              You are strongly advised not to miss these opportunities.

    Grading:  Each of two mid-term exams (tentative dates: September 27 and November 8) 
              will give at most 45 points, the final will give 
              at most 100 points, each of 9 homequizzes (assigned on Tuesdays due 
              the following Tuesday) will give at most 10 points (but only the 
              best six scores will be taken into account), so the maximal 
              possible  score is 250 points. 

              There will be NO CURVING.

Grade computation:  A >= 92 % > A- >= 89 % > B+ >= 86 % > B >= 81 % > B- >= 77 %
                    > C+ >= 74 % > C >= 67 % > C- >= 64 % > D+ >= 61 % > D >= 55 % >F.

    Homework assignments:

1. Chapter 1: Sections 1.1-1.18, Exercises 1-5, pp. 21, 22.      

2. Chapter 2: Sections 2.1-2.14, Exercises 2-4, p. 43
              Sections 2.15-2.20 and 2.23-2.28, Exercises 5,6,8-11, p. 43
              Sections 2.31, 2.32, 2.34-2.42, Exercises 13,14, 16,17, p. 44


Midterm: 
I. Prove that the number ? is not rational.

II. Show that the map ? establishes a 1:1 correspondence between ? and ?.

III. Prove that the set ? is countable (uncountable).

IV. Given a set $E$ in a metric space $X$ find 
the interior, the closure, the boundary of $E$.

V. Something on open and closed sets.

3. Chapter 3: Sections 3.1-3.3, 3.5-3.18, Exercises 1,2,3 p. 78
              Sections 3.21-3.29, 3.33-3.35 
4. Chapter 4: Sections 4.1-4.9, 4.13-4.20, 4.25-4.30, 4.32-4.34
              Exercises 1, 2, 3, 7, 8, 11, 14, 16, 18.

Midterm:
I. Find the limit of a sequence (defined by a formula or recursively)

II. Find the upper limit and the lower limit of a sequence.

III. Show that the function ? is (is not) continuous (uniformly 
continuous).

IV. Specify discontinuities of a function.

5. Chapter 5: Sections 5.1-5.15, Exercises 1, 2, 4, 5, 6, 13, 14

FINAL EXAM.

1. Prove that the number ? is not rational.

2. Show that the number ? is algebraic.

3. Show that the set A is (is not) countable.

4. Establish a one-to-one correspondence between given sets.

5. Something on bounded, open, closed or compact sets.

6. Find limits of sequences defined explicitly or recusively.

7. Show that a given function is (is not) continuous (uniformly 
   continuous) on a given set.

8. Investigate the convergence of a given series.

9. Find supremum (infimum) of a given set.

10. Find upper (lower) subsequential limit of a given sequence.

11. Couple problems on applications of classical theorems:

   2.12, 2.14, 2.41, 2.42, 3.11, 3.14, 3.22, 3.33, 3.34, 4.14,
   4.16, 4.19, 4.23, 5.9, 5.13.