Mathematics 2E: Introduction To Real Analysis

Course Aims
The common thread running through this is the notion of limit. This course will give a precise definition of this notion 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.

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

Main Assessment In: April/May

Course Aims
The common thread running through this is the notion of limit. This course will give a precise definition of this notion 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.
Intended Learning Outcomes of Course
Students should understand and be able to recall the definitions and proofs and be able to apply the results to the types of problem covered in lectures and tutorials.
In particular, students should be able to: deal with implications and equivalences; interpret the negation of a statement involving quantifiers; recognise various methods of proof (direct, contrapositive, counterexample, contradiction, induction); show that a function is bounded/unbounded; show, directly from the definition, that a given number is the limit of a given sequence; evaluate sequence limits using arithmetic and order properties; show that a given sequence is monotonic; investigate sequences defined recursively; use subsequences to establish non-convergence; test series for convergence/divergence; test series for absolute/conditional convergence; determine, directly from the definition, whether a function is continuous; use the sequential characterisation to establish discontinuity;solve problems using the intermediate value and extreme value theorems.