Lecture: TR 14001515 MITCH 113 Vassilev, D.
TA section: W 14001450 MITCH 119 Moraes, J.
Instructor: Dimiter Vassilev Office : Humanities Bldg, Office 447 Email: vassilev@unm.edu Phone Number: 505 277 2136
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: Makeup 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 2773506. In addition, they should see CATS Counseling and
Therapy Services;
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 1^{st} week homework is due on
Wed, Sep 10, together with the 2^{nd} 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 ·
1to1, 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 (bf 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 BolzanoWeierstrass
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 4^{th}) 
Nov. 24 November 27–30 Thanksgiving 
·
Rearrangement
of series 
·
7.4/ 1
(Due December 4^{th}) 
Dec. 1 
·
Root and ratio
tests (for absolute) convergence 11. Continuous functions ·
Subsets of the real
line – limit points, closed, open subsets, HeineBorel
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