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2010/1 Module Catalogue
 Module Code: MAT1003 Module Title: REAL ANALYSIS I
Module Provider: Mathematics Short Name: MS107
Level: HE1 Module Co-ordinator: BRUIN HP Dr (Maths)
Number of credits: 10 Number of ECTS credits: 5
Module Availability
Assessment Pattern

Unit(s) of Assessment
Weighting Towards Module Mark( %)
Two class tests:
Qualifying Condition(s) 
An aggregate mark of 40% is required to pass this module.

Module Overview


Module Aims

The objective of this module is to provide an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module intends to lead to a deeper understanding of what it means when a sequence or series is said to converge. Historic motivation and the rigorous use of definitions and logic play a central role. Tools such as convergence tests are presented and their validity proved.

Learning Outcomes

At the end of the module a student should be able to:

·        demonstrate understanding of the real numbers, their axioms and the role of completeness in the existence of limits and solutions to equations.

·        calculate limits of sequences and (power) series, and prove/disprove converge using the definitions.

·        properly interpret and apply quantifiers in mathematical statement.

  • quote and apply basic theorems in analysis, notably the Theorem of Bolzano-Weierstrass and convergence tests.
Module Content

·        The axioms of real numbers. Denseness of rational and irrational numbers. Maximum, minimum, supremum and infimum of sets, sequences and functions. The triangle inequality.

·        Axiom of Completeness, and its consequence to existence of limits. Role of quantifiers in stating and verifying mathematical definitions.

·        Sequences and convergence, and their properties. Boundedness, Cauchy sequences, subsequences and the Theorem of Bolzano-Weierstrass.

·        Infinite series, convergence and absolute convergence. Convergence tests. Power series, radius and region of convergence. Rearrangement of series.

  • Historical motivation of analysis in mathematics.
Methods of Teaching/Learning

Teaching is by lectures, tutorials and tests. Learning takes place through lectures, tutorials, tests, exercises and background reading.

Autumn semester: 3 lecture/tutorial hours per week for 10 weeks.

Selected Texts/Journals



J. M. Howie, Real Analysis, Springer (2001) 2nd edition, Available in paperback from Springer or




R.P. Burn, Numbers and Functions, Steps into Analysis, Second Edition, Cambridge University Press (2000).
Available in paperback from CUP or



Supplementary Reference Texts (all available in the library)


P.E. Kopp, Analysis, Arnold Publishers, (1990).


K.E. Hirst, Numbers Sequences and Series, Arnold Publishers, (1995).


C. McGregor, J. Nimmo and W. Stothers, Fundamentals of University Mathematics, Albion Publishers, (1994).


M. Spivak, Calculus W.A. Benjamin (1967).

Last Updated

8 September 2010