Your lecture notes will cover all the material
(**except** for those results assigned for self-study) in the syllabus. The occasional chapter references
below are to a more extensive treatment of the topic in question and indicate the ** primary source** of the
material presented in the lectures.

The natural numbers, Peano's axioms, mathematical induction, Peano arithmetic

Aspects of the theory of sets, the axioms of specification and union, De Morgan's laws

Two-fold cartesian products, relations and functions, equivalence relations

The integers: the definition/construction of the set of integers and integer arithmetic, ordering the integers

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

The least upper bound property, the definition/meaning of real line

(The treatment of the above topics follows, although selectively, that of Chapters 1–4 of
Tao's *Analysis 1*)

Dedekind cuts, construction of the real line (Chapter 1: Appendix of Rudin's *Principles*)

The Archimedean property of the real line

Countable and uncountable sets

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

Closure and interior of a set in a metric space, and their relative analogues

Compact metric spaces (Chapter 2 of Rudin's *Principles*)

The characterisation of compact subsets of Euclidean spaces

Properties of compact subsets of a metric space, the Cantor set

Sequences and convergence

Cauchy sequences, the definition of completeness, completeness of the real line

Topics listed up to this point comprise the syllabus of the mid-term examination. **They will
also be a part of the syllabus of the final examination.**

Sufficient conditions for completeness

Subsequences, subsequential limits

The extended real number system, limits at infinity, upper and lower limits (i.e., limsup and liminf)

The limits of special sequences (the section *Some Special Sequences* in Chapter 3 of Rudin's *Principles*)

Infinite series and their convergence, criteria for convergence

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

Absolute convergence, the Ratio and Root Tests,
convergence tests for non-negative series as tests for absolute convergence, conditional convergence

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

Continuous functions

Continuity and compactness, attainment of extreme values, uniform continuity

Connectedness, the characterisation of all connected subsets of the real line

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

Differentition in one variable

The chain rule for differentiation in one variable

Geometric interpretations of the derivative, the relation between critical points and points of local maximum/minimum

Rolle's theorem, Lagrange's mean value theorem and its applications

Higher-order derivatives, Taylor's theorem

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

Riemann integrability of continuous functions, functions with discontinuities

(The treatment of the above topics in the theory of the Riemann integral is that of Chapter 6 of
Rudin's *Principles*, with the following **major** difference: the gauge α in Rudin's
discussion of the Riemann–Stieltjes
integral is just id_{[a, b]} for this course.)

The first and second Fundamental Theorems of Calculus

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

Definition and properties of the logarithm

Sequences of functions, examples and motivations for uniform convergence

Uniform convergence and Riemann integration

The sup-metric, the relationship between uniform convergence and convergence relative to the sup-metric

The space *C*_{b}(*X*;*Y*) and its completeness (when *Y* is complete)

Equicontinuity, compact sets in *C*(*K*;**R**^{n}):
the Arzela–Ascoli Theorem (proof **not** included)

The Weierstrass approximation theorem (Chapter 7 of Rudin's *Principles*.
**Note:** we omit the Stone–Weierstrass theorems.)