Course overview
Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered include: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.
Course learning outcomes
- describe the fundamental properties of the real numbers that underpin the formal development of real analysis;
- Demonstrate an understanding of the theory of sequences and series, continuity, differentiation and integration;
- Demonstrate skills in constructing rigorous mathematical arguments;
- apply the theory in the course to solve a variety of problems at an appropriate level of difficulty;
- demonstrate skills in communicating mathematics.