Real Analysis

The details
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Undergraduate: Level 5
Thursday 05 October 2023
Friday 15 December 2023
03 January 2024


Requisites for this module


MA213, MA302

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 G1F5 Mathematics with Physics (Including Foundation 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 aims of this module are:

  • To introduce the idea of epsilon-delta rigorous analysis.

  • To enhance students’ ability at understanding and writing proofs of results in real analysis.

  • To enhance students’ skills at using results of real analysis.

Module learning outcomes

By the end of the module, students will be expected to:

  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

Indicative 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. Dedekind's axiom and its use in proving that bounded-above, non-empty sets of reals have suprema.
Sequences and convergence. Sums, differences, scalar multiples, products and quotients of convergent sequences.
Cauchy sequences and the equivalence of the Cauchy property and convergence.
Series. Comparison and ratio tests. Absolute convergence implies convergence.
Power series and the radius of convergence.
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.
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.

Learning and teaching methods

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.


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 Coursework weighting
Coursework   Assignment  22/11/2023   
Exam  Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period) 
Exam  Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during January 
Exam  Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period) 

Exam format definitions

  • Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
  • In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
  • In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
  • In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Tao Gao, email:
Dr Tao Gao



External examiner

Prof Stephen Langdon
Brunel University London
Dr Rachel Quinlan
National University of Ireland, Galway
Senior Lecturer in Mathematics
Available via Moodle
Of 30 hours, 27 (90%) hours available to students:
0 hours not recorded due to service coverage or fault;
3 hours not recorded due to opt-out by lecturer(s), module, or event type.


Further information

Disclaimer: The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements, industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules. The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.

The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.