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-han limits (as in the section *Discontinuities* in Chapter 1)