Instructor: Piotr Hajlasz
Office Hours: Monday 2:05-3:00 pm, 4:00-5:00 pm and by appointment.
Office: 420 Thackeray Hall
Email: hajlasz@pitt.edu
Textbook: I will not use my notes Introduction to Analysis. I am currently writing these notes so the file will often be updated.
Intorduction to Analysis. Other sections use the textbook: Jiri Lebl, Basic Analysis. While we will not follow this book, I recommend that you have this text as a suplementary material.
Homework: Posted weekly on the course webpage to be returned to Tian Jing. The problems must be typed in LaTeX. Late homework will not be accepted.
Course Grade: Homework (15%), Quiz (10%), Term Paper(15%), Class midterm (30%), Final (30%).
Course content 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.
Topics
1. Logic, proofs and quantifiers.
2. Basic set theory. Russel's paradox and axioms.
Functions. Cardinality of sets. Cantor-Bernstein theorem.
Uncountability of the set of real numbers.
3. Equivalence relations.
4. Axiomatic definition of the ordered fields and real numbers.
5. Mathematical induction.
6. The Completeness Axiom; Archimedean Property of the real numbers;
density of the rational and irrational numbers in the real numbers.
7. Sequences and an introduction to series; the geometric series;
limits; Limit Laws.
8. The Bolzano-Weierstrass Theorem.
9. Cauchy sequences; Cauchy completeness of the real numbers.
10. Series, convergence tests, alternating series.
11. Limits of functions; continuous functions.
12. Topology of real line.
Homework
HW#1 (LaTeX file) September 11, recitations HW#1 (pdf file)
HW#2 (LaTeX file) September 20, recitations HW#2 (pdf file)
HW#3 (LaTeX file) October 4, recitations HW#3 (pdf file)
HW#4 (LaTeX file) October 18, recitations HW#4 (pdf file)
HW#5 (LaTeX file) November 8, recitations HW#5 (pdf file)