MATH117  METHODS of ANALYSIS


                     
           Instructor: Azer Akhmedov                                                   Teaching Assistant: Peterson Trethewey
           Office:   6702 South Hall                                                          E-mail: peterson@math.ucsb.edu
           E-mail:  akhmedovA.T.math.ucsb.edu                                  MathLab, 1607 South Hall
           Phone:  893 2889                                                                     Thursday 5-7pm.
           Office Hours:   Wed, 11am-2pm.



         TEXTBOOK:  Steven R.Lay, Analysis with an Introduction to Proof.   4th Edition.

         The course covers chapters 3, 4, 5. 
      


         GRADING POLICY:  Homework 40%,  Quiz 30%,  Take-home Final  30%.     


        

                                                               ANNOUNCEMENTS

     1. QUIZ 1 covers sections 10, 11, 12, 13, and 14.

     2. QUIZ 2 covers sections 15, 16, 17, 18, and 19.

     3. QUIZ 3 covers sections 20, 21, 22, 23, and 24.
  

      
       COURSE DESCRIPTION:  Topics include limits, continuity, compactness, connectedness, etc.  The only prerequisite for
                                                         this course is Mathematics 8.  The course intends to introduce students to methods of proof
                                                         in analysis.
                                                 
       HOMEWORK:                      Homeworks are due every Friday in class. Please write neatly, staple your work, and clearly
                                                        print your name on it. Notice that homework weighs 40%! There will be a total of 10
                                                        homeworks during the quarter. The lowest score will be dropped for each student, and the
                                                        highest score will be substituted instead.
          
       QUIZ:                                     3 quizzes will be offered during the quarter. These are intended to test on basic notions to make
                                                        sure that a student has no critical problem in following the course. There will be true/false
                                                        and multiple choice questions in the quizzes. I will post some samples as we get close to it.

       TAKE-HOME FINAL:        Take-home final will involve more proof-based questions. 
                                                       

     

 

 Date 
 Section    
 Topics                                                                
 Homework                                            
Mon 4/3
 Sec.10
 Natural Numbers and Induction
Wed 4/5
 Sec.10
 Natural Numbers and Induction
Fri 4/7
 Sec.11
 Ordered Fields
 HWK#1. Section 10: 
#5,10,13,15,21,22,23
Mon4/10
 Sec.11-12
 Ordered Fields and Completeness
Wed4/12
 Sec.12
 Completeness, Density of Real Numbers
Fri 4/14
 Sec.13
 Topology of Reals
Open and Closed Sets.
 HWK#2.  Sections 11-12:
Sec.11: # 1, 2, 3e,f,g,l. 7*, 10, 11. 
Sec.12: # 1, 2, 3, 6a, 10.
Mon4/17
 Sec.13
 Topology of Reals.
Accumulation Points.
Wed4/19
 Sec.14
 Compactness.
Heine-Borel Theorem.
Fri 4/21
 Sec.15
 Metric Spaces.
 HWK#3. Section 12-13:
Sec.12: # 8.
Sec.13. # 1, 2, 3, 5, 6, 7, 8, 13, 20.
proofs not required for underlined problems, just answers.
Mon4/24
 Sec.16
 Convergance.
Wed4/26
 Sec.16-17
 Convergence and Limit Theorems.
Fri 4/28
QUIZ 1.
 HWK#4. Section 14-15:
Sec.14: # 1, 3, 4, 5, 7
Sec.15: # 1, 2, 3, 8, 9.
Mon 5/1
 Sec.17
 Infinite Limits.
Wed 5/2
 Sec.18
 Monotone Sequences.

Fri 5/3
 Sec.18
 Cauchy Sequences.
 HWK#5. Section 16-17:
Sec.16: #  1, 2, 3, 5, 7, 8c,d, 10, 13.
Sec.17: #  3, 5b,c,g, 6, 8.    
proofs not required for underlined
problems, just answers.
Mon 5/8
 Sec.19.
 Subsequences.
Wed5/10
 Sec.19.
 LimInf and LimSup
Fri 5/12
 Sec.20.
 Limits of Functions.
HWK#6. Section 17-18:
Sec.17: # 1, 2, 7, 15b,c, 18
Sec.18: # 1, 2, 3c,e, 5, 7, 13.
proofs not required for underlined
problems, just answers.
Mon5/15
 Sec.20
 Limits of Functions,
Sequential Criterion
for Limits.
Wed5/17
 Sec.21.
 Continous Functions,
Continuity at a Point
Fri5/19
QUIZ 2.
 HWK#7. Section 19-20:
Sec.19: # 1, 2, 3, 7, 8, 10, 13
Sec.20: # 3, 4, 6, 7.
Mon5/22
 Sec.21.
 Continous Functions.

Wed5/24
 Sec.22
 Properties of Continous Functions, Intermediate Value Theorem.
Fri 5/26
 Sec.22
 Properties of Continuous
Functions, Connectedness.
 HWK#8.  Section 20-21:
Sec.20: # 1, 9, 13, 17.
Sec.21: # 1, 2, 3, 4, 5, 6b-e, 9.  
Mon5/29
 Sec.23.
 MEMORIAL DAY
HAPPY HOLIDAYS!!
Wed5/31
 Sec.23.
 Uniform Continuity
 Handout: Take-home Final
Fri 6/2
 Sec.20-23.
 Uniform Continuity
 HWK#9.  Section 21-22:
Sec.21: # 10, 12, 16, 18.
Sec.22: # 1, 2, 3*, 5, 7, 9, 10.
In all parts of 22.3, if the statement is true, just say it is true, but if it is wrong, give a counterexample.
Solution of 21.16 was given in class.
Mon5/5
 Sec.24.
 Continuity in Metric Spaces.
Wed5/7
 Sec.24
 Continuity in Metric Spaces.
Fri5/9
QUIZ 3.
 HWK#10. Section 23-24.
Section 23. # 1, 2, 3c-f, 5, 6, 11, 15.
Section 24. # 1, 2, 3, 11. 
Mon6/12

Take-home Final Due:
10AM. In the classroom.