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 pth 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 Cb(X;Y) and its completeness (when Y is complete)
Equicontinuity, compact sets in C(K;Rn):
the Arzela–Ascoli Theorem (proof not included)
The Weierstrass approximation theorem (Chapter 7 of Rudin's Principles.
Note: we omit the Stone–Weierstrass theorems.)