# MATH 3A03, Winter 2019

**REAL ANALYSIS I**

The subject matter of this course is the real number system and real-valued functions of real variables. At face value, many of the topics may appear to be the same as those in first year calculus, but the emphasis and goals of the course are completely different. The course focuses on learning to construct rigorous proofs that provide a firm foundation for the kinds of calculations and manipulations learned in elementary calculus courses. We begin by \emph{defining} numbers and consider what properties characterize the real numbers in particular. We re-examine sequences of real numbers and \emph{prove} fundamental results such as the Monotone Convergence Theorem (every bounded monotone sequence of real numbers is convergent). We make the notions of limit and continuity completely rigorous, and generalize familiar concepts such as open and closed intervals to include much more complicated sets. In the process, we reveal many subtleties about real numbers and real-valued functions that form a prelude to many fascinating topics in higher mathematics. The course is required for several of the department's under-graduate programmes, because much advanced mathematics depends critically on a solid understanding of real analysis. The majority of students taking Math 3A03 have taken Math 2R03, Math 2X03 (or ISCI 2A18) and Math 2XX3. However, none of the specific content of these Level II courses is necessary to study real analysis. Students who have done well in first year calculus and linear algebra can request permission to take Math 3A03 in their second year. Students will be evaluated using a set of graded assignments, midterm tests, and a final exam.

**INSTRUCTOR: D. Earn**

**Calendar Description**3 Units

Sequences of real numbers; supremum, continuity. Riemann integral, differentiation. Sequences and series of functions; uniform continuity and uniform convergence.

Three lectures, one tutorial; one term

**Prerequisite(s):**Registration in Level III or above of an Honours program in Mathematics and Statistics; or MATH 2R03 and 2X03; or permission of the instructor

**PLEASE REFER TO MOSAIC FOR THE MOST UP-TO-DATE INFORMATION ON TIMES AND ROOMS**