Real Analysis: Advanced (MAST20033)

This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.

The important distinction between the real numbers and the rational numbers is emphasised and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. Various constructions of the real numbers, for example using Dedekind cuts or by completion, are discussed and shown to be equivalent. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. Compactness of the unit interval is established and various consequences of compactness, such as the Extreme Value Theorem, are discussed. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor’s theorem. We define and compare both the Lebesgue and Riemann integral, establish basic properties of both, and dis- cuss the proof of the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions. Further topics may include: uniform continuity, equicontinuity, the Arzela-Ascoli theorem, and the Stone-Weierstrass theorem.