Instructor. Dr. Marta Lewicka (office hours in Thackeray
408, Monday 3pm - 4pm) Teaching Assistant. Marc Beauchamp. Recitations in
Thackeray 704 from 1:00pm to 1:50pm on Tue and Th.
Prerequisites. The course covers the foundations of
theoretical mathematics and analysis. The principal topics of the
course include fundamentals of logic, sets, functions, number
systems, order completeness of the real numbers and its
consequences, and convergence of sequences and series of real
numbers. Successful completion of Math 0230 (Calculus II) or
equivalent is required to follow this course.
If you do not feel comfortable with the prerequisite material,
please contact the instructor
in the beginning of the course.
Grades.
Your final grade depends on your performance on the final exam as well
as on your total grade. Grades will be based on homework (40%), one
midterm (20%) and the final (40%).
There will be no make up midterm exams. If you miss the midterm exam
for a *documented* medical reason, your grade on it will be the prorated grade of your final exam.
Incompletes will almost never be given,
and only for cases of extreme personal tragedy.
Homework. Homework will be assigned each Thursday
(starting in the second week of the semester),
and it will be due the following Thursday at the recitations. Late homework will not
be accepted. The solution of each exercise will be evaluated in the scale 0-5 points,
taking into account the correctness, clarity and neatness of presentation.
You should solve the problems and write up solutions independently.
Core topics.
1. Logic, proofs and quantifiers. Basic set
theory. Functions. Equivalence relations.
2. Elementary properties of the natural numbers; mathematical
induction.
3. Axiomatic introduction to the ordered fields of rational and real
numbers.
4. Elementary inequalities.
5. The Completeness Axiom; Archimedean Property of the real numbers;
density of the rational and irrational numbers in the real numbers.
6. Countability of the rationals; decimal expansions of real numbers;
uncountability of the real numbers.
7. Sequences and an introduction to series; the geometric series;
limits; Limit Laws.
8. The Monotone Convergence Theorem.
9. The Bolzano-Weierstrass Theorem.
10. Cauchy sequences; Cauchy completeness of the real numbers.
11. Series; convergence tests; alternating series; conditional
convergence and rearrangements.
12. Cluster points; limits of functions; continuous functions and
examples of sets of points of discontinuity.
Calendar.
4 Jan (Wed): First class
16 Jan (Mon): No class (Martin Luther King's day)
6, 8, 10 Mar (Mon, Wed, Fri): Spring break - no classes
21 Apr (Fri): Last class
29 Apr (Saturday) 8:00AM - 9:50AM: Final Exam (in Thack 704)