Introduction to Real Analysis

Course Description

An investigation into the mathematical foundations of calculus.  Through lectures and exercises, visit fundamental concepts of mathematical analysis including logic, basic set theory, functions, number systems, order completeness of the real numbers and its consequences, sequences and series, topology of R^n, continuous functions, uniform convergence, compactness, and theory of differentiation and integration.  Expand mathematical proof and writing skills through ample practice with LaTex to communicate mathematics effectively and demonstrate rigorous math thinking in preparation for more advanced courses.

Course Contents

Basic Set Theory and Mathematical Logic
Definition and properties of Fields
Real number system
Fundamental Property of real numbers
Sequence and Limits
Properties of limits, bounded and monotone sequences
Bolzano-Weierstrass Theorem and Cauchy sequence
Series and convergence test
Basic topology of real line and limits of functions
Limits and continuity of functions
Continuous function on compact interval and uniform continuity
Derivatives and Mean Value Property
Riemann Integral and Fundamental Theorem of Calculus
Metric spaces introduction


Exam 1 : 30% , Exam 2: 30%, Homework: 40%

Prerequisites or Prior Knowledge

Successful completion of undergraduate Calculus or equivalent courses is required to take this course. Multivariable calculus is not a prerequisite. If you are not sure about the prerequisite material, please contact the instructor at the beginning of the course.


Introduction to Real Analysis, Robert G. Bartle and Donald R. Sherbert, 4th edition.

Reference Books

Principles of Mathematical Analysis, Rudin, 3rd edition.


Alternate years course, AY2024

Research Specialties