This course is a first introduction to real analysis. The common thread running through the course is the notion of limit. The precise definition of this notion will be given for both sequences and series. It is an essential course for intending honours students. The emphasis is on developing and applying standard techniques of proof to give rigorous arguments from basic definitions.

2 x lectures per week. Fortnightly tutorials

One degree examination (80%) (1 hour 30 mins); coursework (20%).

Reassessment will not be available for continuous assessment.  Immediate feedback will be given to students at the assignment deadlines, and it is not practical to set multiple versions of each of these exercises.

In total 80% of the assessment can be repeated for students not reaching the threshold grade. 

Note that students are required to submit 75% of the assessments in order to receive a grade for the course. Accordingly, students who have a substantial period of illness and have all eAssignments and feedback exercises MV'ed, will not be in danger of failing to receive credits.

This course is a first introduction to real analysis. The common thread running through the course is the notion of limit. The precise definition of this notion will be given for both sequences and series. The notion of continuity for functions will be discussed and related to convergence of sequences. Some important consequences of continuity to be studied are the intermediate value theorem and its applications, and the existence of extrema. The emphasis is on developing and applying standard techniques of proof to give rigorous arguments from basic definitions.

By the end of this course, students will be able to:

■ Negate statements involving logical quantifiers; given a conditional statement containing and, or, not, and logical quantifiers: determine logically equivalent statements, identify its hypothesis and conclusion, and determine its converse and contrapositive; recognise various methods of proof (for example, direct, contrapositive, counter example, contradiction, induction).

■ Show that a function is bounded or unbounded; prove, from first principles, that a given number is the limit of a given sequence; evaluate sequence limits using arithmetic and order properties.

■ Given a sequence, including those defined recursively, prove that it is monotonic (or not); use sub-sequences to establish non-convergence; determine if it is convergent or divergent; determine if it is absolutely or conditionally convergent.

■ Determine, from first principles, whether a function is continuous (or not); apply the sequential characterisation to establish discontinuity; solve problems using the intermediate value and extreme value theorems.

■ State all definitions and results (lemmata, theorems, corollaries, propositions) covered in lectures, and apply these to solve suitable problems.

■ Prove the results covered in lectures, apply and adapt these proofs in novel situations.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.