**Time and Place:**MWF 11:30am–12:20pm in REC 113

**Instructor:**Arshak Petrosyan

**Office Hours:**MWF 9:30-10:30, or by appointment, in MATH 610

**Textbook:**[R] W. Rudin,

*Principles of mathematical analysis*, Third edition, McGraw-Hill, New York, 1976

**Homework**will be collected weekly on Wednesdays. The assignments will be posted on this website at least one week prior the due date.

**Exams:**There well be two midterm exams (evening exams) and a comprehensive final exam (covering all material). The exact time and place will be specified at least two weeks in advance.

**Description:**Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem.

**Syllabus:**

Chapter 1. The Real and Complex Number System

- Real number system - (Emphasize inf, sup)
- Extended real number system
- Euclidean spaces

- Finite, countable and uncountable sets
- Metric spaces (Only a few special examples)
- Compact sets

- Convergent sequences
- Subsequences
- Cauchy sequences
- $\limsup x_n$ and $\liminf x_n$
- Series
- Series with many terms (comparison test)
- Absolute and conditional convergence
- Rearrangements

- Limits of functions
- Continuous functions
- Continuity and compactness
- Intermediate Value Theorem

- Definition and existence
- Properties
- Integration and differentiation

- Uniform convergence
- Uniform convergence and continuity
- Uniform convergence and integration
- Uniform convergence and differentiation
- Equicontinuous families of functions
- Stone-Weierstrass Theorem

- Sets of Lebesgue measure zero
- Characterization of Riemann integrable functions bounded and continuous a.e.
- Differentiability a.e. of monotone functions