Intro Real Analysis

Lecture: TR 1400-1515 MITCH 113 Vassilev, D.

TA section: W 1400-1450 MITCH 119 Moraes, J.

                                                                 

Instructor: Dimiter Vassilev     Office :  Humanities Bldg, Office 447  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: TTh 10:00-11:00, 13:30-14:00  or by appointment                                                                                                                           

 

Final Exam: Tuesday, December 16, 10:00am–12:00 p.m.

 

Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, November 14, 2008. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before November 14, 2008, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses.           

Text: Terence Tao, Analysis I.

Please note the following guidelines for the course:

 

Description: Rigorous treatment of calculus in one variable. Definition and topology of real numbers, sequences, limits, functions, continuity, differentiation and integration. Students will learn how to read, understand, and construct mathematical proofs.

 

Prerequisite: Math 264 and two courses at the 300+ level.


Grades: The final grade will be determined by homework (25%), two midterms( 50%) and a final exam (25%). 

 

Homework:  Homework is due every Wednesday. There will be one HW  weekly. You can work together on the homework, but you do need to write up your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly (no points for work that the reader cannot follow- this is true for exams), and staple the sheets.   The lowest homework grades will be dropped. Please no late homework

 

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given. 

 

Disability Statement: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs and they should also contact Accessibility Services in Mesa Vista Hall, Room 2021, phone 277-3506.  In addition, they should see CATS- Counseling and Therapy Services; Student Health Center (277-4537). (They can help if you suffer from exam anxiety).    

 

 

                                             Syllabus – Fall  2008  (please check after class as advanced postings of homework could change)

 Week of

Topics Covered

·    Due Homework – turn in Wednesday following week in discussion section.

·    (Note 3.1/ 2 means problem 3.1.2, i.e. problem #2 from section 3.1)

Aug. 25

1.      Natural numbers Ch. 2

·        Peano’s axioms

·        Addition – commutative, associative, cancellation law;

·        Ordering of the natural numbers – properties.

 

·    p. 33/1, 2, 3

·    Note: The 1st week homework is due on Wed, Sep 10, together with the 2nd week homework.

Sep. 1

·        Multiplication of natural number.

 

2. Axioms of set theory

3. Functions – basic definitions

·    p. 36/3, 5    p. 51/ 6, 8, 11

·    p.54 3.2/2*

Sep. 8

      4. Functions

·        1-to-1, bijective,

·        image of a set

5.      More set axioms

·        power set, union;

·        Cartesian products of sets.

·    3.3/ 1, 2, 4, 7, 8

·    3.4/ 9, 11

 

Sep. 15

·        Cartesian products of sets , axiom of choice;

     6. The integer numbers

·    3.5/ 3, 10  8.4/ 1 3.6/ 5, 9

·    4.1/ 1

Sep. 22

·        Properties

     7. The rational numbers

·    4.1/ 1, 3, 6, 7 (b-f parts of the lemma)

·    4.2/2, 4, 6

Sep. 29

· Absolute value and exponential

· “Gaps” in the rational numbers

·    4.3/4, 5

·    4.4/1, 2, 3

Oct. 6

8. The real numbers

·        Cauchy sequences

·        The construction of the real numbers

 

·    5.2/ 1, 2  5.3/2, 3. (due Tuesday Oct. 21)

Oct. 13

October 16–17 Fall Break

· The construction of the real numbers

·    5.4/1  (due Tuesday Oct. 21)

Oct. 20

·        Questions

·        Exam #1 Thursday, October 23

·     

Oct. 27

·       Ordering the real numbers. The rational numbers are dense.

·       The least upper bound

·    5.4/,3, 4, 5, 7

·    5.5/ 1, 4

Nov. 3

 

·      Real exponentiation

 

9. Limits of sequences

·        Convergence and some standard limits

·        Limit points of a sequence, limsup and liminf.

·    5.6/ 2, 3, problem assigned in class

·    6.1/ 1, 5 , 19c), d), h)

Nov. 10

·         The Bolzano-Weierstrass theorem

·        The exponential function

 

 

·    6.3/ 4    6.4/ 7, 8, 9  6.5/ 2   6.6/3

 

Nov. 17

10. Series

·        Absolute and conditional convergence

·        Geometric series, alternating series

 

 

·    6.7/ 1  7.2/ 1, 4, 6 7.3/ 2 (Due December 4th)

Nov. 24

November 27–30 Thanksgiving

 

·        Rearrangement of series

 

·    7.4/ 1 (Due December 4th)

 

Dec. 1

·        Root and ratio tests (for absolute) convergence

11. Continuous functions

·        Subsets of the real line – limit points, closed, open subsets, Heine-Borel theorem

 

Exam #2

·    7.5/ 2, 3   9.1/ 4, 9, 13   9.3/ 3

·    9.4/ 2, 7  9.6/ 1  9.7/ 1, 2

Dec. 8

12. Differentiation

 

 

·         

·         

·     

Dec. 15

Final Exam: Tuesday, December 16, 10:00am–12:00 p.m.    

·     

 

 

 

Labor Day Holiday (Saturday classes meet as scheduled) (no classes/University closed) . September 1, 2008

Fall Break (Saturday classes meet as scheduled) (no classes) . October 16–17, 2008

Thanksgiving Holiday (no classes/University closed) . November 27–30, 2008

Final examinations December 15–20, 2008