Your lecture notes will cover all the material
(except for those results assigned for self-study) in the syllabus. The occasional chapter references
are to more extensive explanations, and refer to Rudin's *Principles*.

Aspects of the theory of sets, relations and functions

The natural numbers, the principle of mathematical induction, Peano arithmetic

Number systems, the rational numbers, fields, ordered fields and the "usual order" on the rationals

The least upper bound property, the real line, construction of the real line (Chapter 1: Appendix)

The Archimedean property of the real line, complex numbers, Euclidean spaces, the Cauchy–Schwarz inequality
and associated inequalities (the section *The Complex Field* in Chapter 1)

Countable and uncountable sets, cardinality

Metric spaces, open and closed sets in metric spaces and associated concepts

Compact sets, the characterisation of compact subsets of Euclidean spaces

Cantor sets, perfect sets (the section *Perfect Sets* in Chapter 2), connected sets

Sequences and convergence

Subsequences, subsequential limits, the limits of special sequences (the section *Some Special Sequences* in Chapter 3)

Cauchy sequences, completeness, sufficient conditions for completeness

Infinite series and their convergence, criteria for convergence

Convergence tests for non-negative series,
criterion for summability of the series of *p*th powers

The extended real number system, limits at infinity, upper and lower limits

Absolute convergence, the Ratio and Root Tests, conditional convergence,
using convergence of series to derive limits of sequences

Topics listed up to this point comprised the syllabus of the mid-term examination.

The limit of a function: various **equivalent** definitions, the algebra of limits

Continuous functions

Continuity and compactness, attainment of extreme values, uniform continuity

Continuity and connectedness, the intermediate-value theorem, and applications

Left- and right-hand limits (as in the section *Discontinuities* in Chapter 1)

Differentition in one variable

Lagrange's mean value theorem and its applications

The chain rule for differentiation in one variable

Taylor's theorems: the approximation theorem and the mean value theorem

Integration: motivation, the Riemann integral, and a characterisation of Riemann integrability

Riemann integrability of continuous functions, functions with discontinuities

The first and second Fundamental Theorems of Calculus

Techinques: the change-of-variable formulas for the Riemann integral, integration by parts

Sequences of functions, examples and motivations for uniform convergence

Uniform convergence and Riemann integration

The normed linear spaces *C*_{b}(*X*;**R**) and *C*_{b}(*X*;**C**) and their completeness

Equicontinuity, compact sets in *C*(*K*;**R**): the Arzela–Ascoli Theorem

The Weierstrass approximation theorem (**Note:** we omit the Stone–Weierstrass theorems.)

Vector-valued functions in several real variables: limits and continuity

Matrix representations of linear transformations, the matrix norm

The meaning of differentiation of functions in several variables: the total derivative

Partial derivatives, matrix representation of the total derivative

The chain rule for the total derivative, applications of the chain rule