# Mathematics 2E: Introduction To Real Analysis MATHS2007

• School: School of Mathematics and Statistics
• Credits: 10
• Level: Level 2 (SCQF level 8)
• Typically Offered: Semester 2
• Available to Visiting Students: Yes
• Available to Erasmus Students: Yes

#### Short Description

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.

#### Timetable

Lectures on Tuesdays and Thursdays at 10.00 am or Tuesdays and Thursdays at 12.00 noon. Fortnightly seminars on Mondays.

#### Requirements of Entry

Mathematics 1R or 1X at grade D and 1S or 1T or 1Y at grade D and a pass in the level 1 Skills test.

Mathematics 2U

#### 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.

#### Minimum Requirement for Award of Credits

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