MATH 0413 Fall 2011

Schedule and Contact Information
Classes: M W F 11:00 - 11:50, Thackeray 627
Instructor: Alexander Borisov
Office: Thackeray 414
Email: borisov"at"pitt"dot"edu
Recitations: Tu Th 11:00 - 11:50, Thackeray 525
Recitation Instructor: Aziz Takhirov

Announcements and Homework
Please refer to the List of Mini-quizzes for the most updated list of definitions and statements of theorems that can be asked on a mini-quiz.
Please refer to the List of Homework Assignments for the written homework assignments and due dates.

Prerequisites
Calculus II or equivalent, and some natural curiosity.

Text
We are using the free online book by Jiri Lebl, Basic Analysis, with Pitt supplements by Frank Beatrous and Yibiao Pan. Please let me know if you are interested in purchasing a hard copy for \$16-\$18.

Overview
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

• 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 Bolzano-Weierstrass Theorem.
• 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. You will also learn to recognize and correct some mistakes in mathematical proofs.

Notes on the Structure of the Course
1. Recitations are an indispenable part of the course. Attendance and appropriate participation are required, part of the course grade comes from the recitation grade, which will be assigned by your recitation instructor.
2. In the beginning of every lecture you will be asked to answer 2-5 simple questions on the definitions and statements of theorems. The primary purpose of these mini-quizzes is to ensure that you can follow the topic of each lecture.
3. In addition to the daily mini-quizzes, you will have regular quizzes (during recitations, announced in advance), and homework. Their purpose is to help you achieve the mastery of the material required for solving problems on the midterm and the final, and to improve your mathematical writing skills. Some of these quizzes and homework may specifically address mathematical writing and proof comprehension/analysis.