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Real Analysis - MAT00005C

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  • Department: Mathematics
  • Module co-ordinator: Dr. Simon Eveson
  • Credit value: 20 credits
  • Credit level: C
  • Academic year of delivery: 2021-22
    • See module specification for other years: 2022-23

Module will run

Occurrence Teaching period
A Spring Term 2021-22 to Summer Term 2021-22

Module aims

The first years of all mathematics programmes are designed to give students a thorough grounding in a wide spectrum of mathematical ideas, techniques and tools in order to equip them for the later stages of their course. During first year, as well as consolidating, broadening and extending core material from pre-University study, we initiate a cultural transition to the rigorous development of mathematics which is characteristic at University. Students will develop both their knowledge of mathematics as a subject and their reasoning and communication skills, through lectures, tutorials, seminars, guided self-study, independent learning and project work. This development is addressed in all of our first year modules, although different modules have a different emphasis.

In addition to the above broad aims of the first year, this module focusses on the description and study of the characteristic properties of the real number system, the mathematical concept of limit, the formal definitions of continuous, differentiable, and integrable real-valued functions of a real variable. In this module students will be introduced to a rigorously developed mathematical theory and will be provided with the foundations for a range of higher level mathematics modules which involve the basic concepts of analysis, available in Stages 2–4.

Module learning outcomes

Subject content

  • Real numbers. The real number system, including the existence of least upper and greatest lower bounds, the property of Archimedes, well-ordering of the integers, rational density, existence of q-th roots of positive numbers, definition of rational powers of positive numbers.
  • Sequences. Limits of sequences, algebra of limits, Sandwich Theorem, standard limits. Principle of bounded monotone convergence, subsequences, the Bolzano-Weierstrass Theorem and convergence of Cauchy sequences. 
  • Series. Examples including the harmonic and geometric series. Absolute and conditional convergence, convergence tests (e.g. Leibniz test, Cauchy condensation test, comparison test, ratio test). Real power series and the radius of convergence. Exponential and trigonometric functions as power series.
  • Continuity. Limit of a real function at a point (both epsilonic and sequential versions), algebra of limits, continuity. The Intermediate Value Theorem. A continuous function on a closed bounded interval attains its extrema.
  • Differentiation. The derivative as a limit. Algebra of derivatives. Differentiability of polynomials and power series.  Rolle's theorem and the Mean Value Theorem. Inverse Function Theorem.
  • Riemann integral. Definition of the Riemann integral, basic properties (linearity, domain decomposition), integrability of continuous functions. Fundamental Theorem of Calculus. Integration of power series.

Academic and graduate skills

  • Academic skills: the application of rigorous mathematical techniques and ideas to the development of mathematics; the power of abstraction as a way of solving many similar problems at the same time; the development and consolidation of essential skills which a mathematician needs in their toolkit and needs to be able to use without pausing for thought.

  • Many of the techniques and ideas developed in this module are ones which graduates employed as mathematicians and in other numerate professions will use from day to day in their work. On top of this, students, through lectures, examples, classes, will develop their ability to assimilate, process and engage with new material quickly and efficiently.


Task Length % of module mark
Coursework - extensions not feasible/practicable
N/A 5
Online Exam -less than 24hrs (Centrally scheduled)
Real Analysis
3 hours 95

Special assessment rules



Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Re-assessment: Real Analysis exam
3 hours 100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

M Hart, A Guide to Analysis, Palgrave.

G B Thomas, M D Weir, J Hass, F R Giordano, Thomas' Calculus (11th edition), Pearson, 2004.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.