MATH 0413 Fall 2010

Schedule and Contact Information
Lecture: M W F 1:00 - 1:50, Cathedral of Learning 226
Recitation: Tu Th 1:00 - 1:50, Thackeray 524
Instructor: Alexander Borisov
Office: Thackeray 414
Email: borisovatpittdotedu
Office Hours: by appointment

Announcements and Homework
Quiz on Theorems will be on Monday, Dec. 6th. Theorems will have to be stated, not proven. They will be chosen from the list of main theorems. This is also your chance to redo any missed definitions.
Definitions Quiz 3 will be on Monday, November 15th: Set 2 of the updated list of definitions. This is also your chance to redo the missinf definitions from Set 1.
Midterm date has been set: Wednesday, October 20th. Click here for the Study Guide.
Regular Homework: None assigned at this time.
USAMTS Homework: assignment 1 is due Friday, 10/15: Year 10, Round 1, problems 1--5.
Approximate Day-by-day Lecture Description.


Two terms of calculus on the level of Math 0220 and 0230.


Bartle and Sherbert, Introduction to Real Analysis, Third Edition, Wiley

The course covers Chapters 1 through 3.


The course covers the foundations of theoretical mathematics and analysis. Topics include sets, functions, number systems, order completeness of the real numbers and its consequences, and convergence of sequences and series of real numbers.

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.

Course Goals

This course is a prerequisite for MATH 0420, and together they provide a rigorous foundation of the one-variable Calculus. Besides learning the topics listed above, you will get a better understanding of what constitutes a rigorous mathematical proof. This is a writing-intensive course, and during the semester you will be writing mathematical proofs of increasing degree of complexity.


Midterm (Date to be announced)25%
Final (Date and place to be announced)  35%

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