In most Calculus courses, we learned many useful computation techniques without further explanation of the concepts behind the tools. Real analysis, which sometimes can be roughly understood as “advanced calculus”, is to set the solid mathematical foundation for calculus. In this course, we will visit many fundamental concepts of mathematical analysis through the lecture and exercises.

The principal topics of the course include fundamentals of 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, theory of differentiation and integration.

**Target students and prerequisites:**

The course is an introductory course and is designed to be accessible to students that are seeing proofs for the first time. The only prerequisite is an understanding of the results from single-variable calculus. 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.

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

Successful completion of undergraduate single-variable calculus