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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

Mathematics 1 at at least grade D3.

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

It is the default expectation that all courses will offer opportunities for reassessment or deferred assessment. Where it is not possible to offer this in some assessment components, the grade achieved at the first attempt will be counted towards the final course grade, and any exceptions for this course are described below.

[No exceptions]

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.