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 2:00-2:45 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.


Some level of mathematical maturity is required. This is a fast-paced course for the highly motivated students.


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 writing-intensive 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 5-10 minute mini-quiz on definitions and/or statements of theorems. Check here for the periodically updated list of past mini-quizzes.

Core topics

  1. Logic, proofs and quantifiers. Basic set theory. Functions and 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. Real-valued functions on an interval: limits and continuity.
  12. Intermediate Value Theorem; Max-Min Theorem.
  13. Uniform continuity; continuous functions on a closed and bounded interval are uniformly continuous.
  14. Differentiable functions.
  15. Interior Extremum Theorem, Rolle's Theorem, Mean Value Theorem.
  16. Taylor's Theorem and Taylor Series.
  17. The Riemann Integral on a closed and bounded interval. Increasing functions are Riemann-integrable. Continuous functions are Riemann-integrable.
  18. 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) 624-7890 as early as possible in the term.

Helpful Links

USA Mathematical Talent Search (USAMTS) Problems and Solutions