MATH 0450
Spring 2011
Schedule and Contact Information
Lecture: M W F 1:00  1:50, Thackeray 627
Recitation: Tu Th 1:00  1:50, Thackeray 524
Instructor: Alexander Borisov
Office: Thackeray 414
Email: borisovatpittdotedu
Office Hours: MW 2:002:45 pm + by appointment
Recitation instructor: Mark Tronzo
News and Announcements
List of Theorems for the Final Exam.
Sample Problems for the Final Exam.
Homework due Friday, 2/4: Section 2.1. 10,11,12,18,20.
USAMTS Homework due Friday, 3/4: Year 10, round 3.
Prerequisites
Some level of mathematical maturity is required. This is a fastpaced course for the highly motivated students.
Text
Bartle and Sherbert, Introduction to Real Analysis, Third Edition,
Wiley
There are many other good Analysis textbooks and web resources. In particular, you may find useful professor Piotr Hajlasz's Introduction to Analysis (based on MATH0450, Spring 2010 notes, courtesy of Piotr Hajlasz).
Course Description
This course provides an introduction to the theoretical aspects of mathematical analysis for highly motivated students, and will replace Math 0413 and 0420 for those students. It serves as an entry point for students contemplating an honors major in Mathematics. Topics include sets and functions, number systems, topology of the real line, limits, continuity, and the main theorems of elementary calculus. Before registering for a UHC course, students must obtain special permission from the University Honors College, 3600 Cathedral of Learning.
This is a writingintensive course. During the semester you will be writing mathematical proofs of increasing degree of complexity. For this purpose we will use the course material and/or the "elementary" mathematics problems from the
USA Mathematical Talent Search (USAMTS).
Tests, Quizzes and Homework
The grade in the course will be determined primarily by the grades on the final examination and one or more midterm examinations. Additionally, there will be graded homework and quizzes that will affect the grade in the course. By far the most important homework is to understand the material covered in lectures. If you have to miss a lecture, it is your responsibility to get notes from another student and study them as soon as possible. You should also notify your instructor and seek help from the instructor or the recitation instructor as needed. Most topics in the course rely heavily on the previously covered material, and the pace is twice as fast compared to Math 0413 or Math 0420. So if you find yourself behind, seek help immediately.
Every class will start with a 510 minute miniquiz on definitions and/or statements of theorems. Check here for the periodically updated list of past miniquizzes.
Core topics

Logic, proofs and quantifiers. Basic set theory. Functions and relations.

Elementary properties of the natural numbers; mathematical induction.

Axiomatic introduction to the ordered fields of rational and real numbers.

Elementary inequalities.

The Completeness Axiom; Archimedean Property of the real numbers; density
of the rational and irrational numbers in the real numbers.

Countability of the rationals; decimal expansions of real numbers;
uncountability of the real numbers.

Sequences and an introduction to series; the geometric series; limits;
Limit Laws.

The Monotone Convergence Theorem.

The BolzanoWeierstrass Theorem.

Cauchy sequences; Cauchy completeness of the real numbers.
Realvalued functions on an interval: limits and continuity.

Intermediate Value Theorem; MaxMin Theorem.

Uniform continuity; continuous functions on a closed and bounded
interval are uniformly continuous.

Differentiable functions.

Interior Extremum Theorem, Rolle's Theorem, Mean Value Theorem.

Taylor's Theorem and Taylor Series.

The Riemann Integral on a closed and bounded interval.
Increasing functions are Riemannintegrable. Continuous functions are
Riemannintegrable.

The Fundamental Theorem of Calculus.
Academic Integrity
Cheating/plagiarism will not be tolerated.
Violations of the
School of Arts and Sciences Policy on Academic Integrity
will result in a minimum sanction of a zero score for
the quiz, exam or paper in question. Additional sanctions may be imposed,
depending on the severity of the infraction.
Homework Policy
You may work with other students or use library resources, but each student
must write up his or her solutions independently. Copying solutions from other
students will be considered cheating, and handled accordingly.
Disability Resource Services
If you have a disability for which you are or may be requesting an
accommodation, you are encouraged to contact both your instructor and the
Office of Disability Resources and Services,
216 William Pitt Union (412) 6247890 as early as possible in the term.
Helpful Links
USA Mathematical Talent Search (USAMTS) Problems and Solutions