# Module Directory

## MA203-5-SP-CO:Real Analysis 1

The details
2020/21
Mathematical Sciences
Colchester Campus
Spring
Current
Sunday 17 January 2021
Friday 26 March 2021
15
15 July 2020

Requisites for this module
MA118
(none)
(none)
(none)

MA213

## Key module for

BSC L1G2 Economics and Mathematics (Including Placement Year),
BSC LG11 Economics and Mathematics,
BSC LG18 Economics and Mathematics (Including Foundation Year),
BSC LG1C Economics and Mathematics (Including Year Abroad),
BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
MMATG198 Mathematics,
BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year),
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad),
MSCIG199 Mathematics and Data Science

## Module description

This is an introductory epsilon-delta analysis module. Students will develop their sense of rigour and precision.

## Module aims

The module aims to:

- introduce the idea of epsilon-delta rigorous analysis.
- enhance studentsâ€™ ability at understanding and writing proofs of results in real analysis.
- enhance studentsâ€™ skills at using results of real analysis.

## Module learning outcomes

On completion of the module students should:

1. Understand basic proofs and proof techniques in relation to real numbers, suprema and infima, limits (sequences, series and functions), continuity and differentiability
2. Be able to give proofs of some simple standard facts in rigorous real analysis
3. Be able to use these techniques on appropriate problems, including working out proofs of simple results related to the module.

## Module information

Syllabus

Numbers systems such as the real and rational numbers. Basic properties of real numbers: field structure, order relation, triangle inequality, Archimedes' axiom. (No formal construction of the real numbers). Suprema and infima, and examples. Dedekind's axiom and its use in proving that bounded above non-empty sets of reals have suprema.
Sequences. Convergence of sequences. Multiple examples. Sums, differences, scalar multiples, products and quotients of convergent sequences.
Cauchy sequences and the equivalence of the Cauchy property and convergence
Series. Comparison aad ratio tests. Absolute convergence implies convergence.
Power series and their radii of convergence. Examples.
Limits of functions
Continuous functions of one real variable; related results such as sums, products, quotients and compositions (chain rule).
Intermediate Value Theorem. Boundedness, and attainment of bounds, for continuous functions on closed bounded intervals. Relevant examples.
Differentiable functions. Examples of differentiable and non-differentiable functions. Differentiable implies continuous.
Theorems related to continuous and differentiable functions, such as Rolle's Theorem and the Mean Value Theorem. Relevant examples.

## Learning and teaching methods

Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support. Students experiencing difficulties with the module are encouraged to talk to the lecturer during office hours/open-door policy.

## Bibliography*

This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

## Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework   Test
Exam  120 minutes during Summer (Main Period) (Main)

Coursework Exam
20% 80%

### Reassessment

Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Alastair Litterick, email: a.litterick@essex.ac.uk.
Dr Alastair Litterick & Dr David Penman
Dr Alastair Litterick (a.litterick@essex.ac.uk), Dr David Penman (dbpenman@essex.ac.uk)

Availability
Yes
Yes
No

## External examiner

Dr Tania Clare Dunning
The University of Kent
Resources
Available via Moodle
No lecture recording information available for this module.

Further information
Mathematical Sciences

* Please note: due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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