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
The rational numbers: the definition/construction of the set of rationals and rational arithmetic, the rationals as a field
Ordered sets, the "usual order" on the rationals, ordered fields, the least upper bound property
(The treatment of the above topics follows, although selectively, that of Chapters 1–4 of
Tao's Analysis 1)
The least upper bound property, the definition/meaning of the system of real numbers
Dedekind cuts, construction of the real line (Chapter 1: Appendix of Rudin's Principles)
The Archimedean property of the real line
Metric spaces, open and closed sets in metric spaces and associated concepts, the closure of a set
Open and closed sets relative to a metric subspace
Compact sets in a metric space (Chapter 2 of Rudin's Principles)
The characterisation of compact subsets of Euclidean spaces
Countable and uncountable sets
Sequences and convergence
Subsequences, subsequential limits
Extracting convergent subsequences and the role of compactness
Cauchy sequences, the definition of completeness
Sufficient conditions for completeness
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.
The extended real number system, limits at infinity, upper and lower limits (i.e., limsup and liminf)
Self-study: The limits of special sequences (the section Some Special Sequences in Chapter 3 of Rudin's
Principles)
Infinite series and their convergence, the Cauchy criterion
Convergence tests for non-negative series, the Ratio and Root Tests,
Cauchy's Condensation Test
Absolute convergence, conditional convergence
The Ratio and Root Tests as tests for absolute convergence, power series
The limit of a function: various equivalent definitions, the algebra of limits
Continuity at a point, continuous functions
Continuity and compactness, attainment of global maximum/minumum values, uniform continuity
Connectedness, the characterisation of all connected subsets of the real line
Continuity and connectedness, the Intermediate Value Theorem
Differentition in one variable
Review (self-study): The algebra of derivatives, the chain rule for differentiation, the relation between critical
points and points of local maximum/minimum, Rolle's theorem, Lagrange's mean value theorem and its applications
(pages 104–108 of Rudin's Principles)
Higher-order derivatives, Taylor's theorem
Integration: motivation, the Riemann integral, and a characterisation of Riemann integrability
Riemann integrability of continuous functions
(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.)
Review (self-study): The first Fundamental Theorem of Calculus
Primitives/antiderivatives, the second Fundamental Theorem of Calculus
Techinques of integration: integration by parts (the change-of-variables
theorem, which you know from UMA101, will be of no relevance in our present treatment of integration)
Sequences of functions, motivations for uniform convergence, and some examples
Uniform convergence and Riemann integration
The sup-metric, the relationship between uniform convergence and convergence relative to the sup-metric
The space Cb(X;Fk), the Weierstrass Approxination Theorem (only its statement)
Equicontinuous families,
the Arzela–Ascoli theorems (note: proofs not included), the interpretation of the Arzela–Ascoli Theorem in terms of compact sets in
C(K;F)