Last modified: 31 Jul 2023 11:19
Analysis provides the rigorous, foundational underpinnings of calculus. It is centred around the notion of limits: convergence within the real numbers. Related ideas, such as infinite sums (a.k.a. series) and continuity are also visited in this course.
Care is needed to properly use the delicate formal concept of limits. At the same time, limits are often intuitive, and we aim to reconcile this intuition with correct mathematical reasoning. The emphasis throughout this course is on rigorous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.
|First Sub Session
|15 credits (7.5 ECTS credits)
- Fundamental properties of real numbers: field operations, order, completeness.
- Sequences and limits: convergence, basic examples, methods of deducing convergence, properties of convergent sequences, the Bolzano-Weierstrass Theorem.
- Infinite sums (series): convergence, convergence tests.
- Functions of one real variable: limits and continuity, methods of deducing limits, Extreme Value Theorem, Intermediate Value Theorem, uniform continuity.
To put on a sound footing many of the results, procedures, and concepts used in Calculus. It will include a discussion of fundamental properties of real numbers, sequences and limits, series, and continuity of functions. Some applications will also be given.
Information on contact teaching time is available from the course guide.
There are no assessments for this course.
Best of (resit exam mark) or (resit exam mark with carried forward CA marks)
|Be able to use the theorems of the course in unseen situations;
|Have developed the ability to prove elementary results, and be able to detect fallacious arguments;
|Be able to state the main definitions and theorems of the course;
|know about basic properties of the real numbers and what distinguishes them from the rational numbers;
|Be able to establish the convergence of simple sequences and series
|be familiar with the concepts of limits and continuity.
|know precise definitions and basic properties of elementary functions;