Module Code: MAT1003 
Module Title: REAL ANALYSIS I 

Module Provider: Mathematics

Short Name: MS107

Level: HE1

Module Coordinator: BRUIN HP Dr (Maths)

Number of credits: 10

Number of ECTS credits: 5




Module Availability 
Autumn 


Assessment Pattern 
Unit(s) of Assessment

Weighting Towards Module Mark( %)

Two class tests:
Exam:

60
40

Qualifying Condition(s)
An aggregate mark of 40% is required to pass this module.




Module Overview 



Prerequisites/Corequisites 
None. 


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.



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



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 
Essential
J. M. Howie, Real Analysis, Springer (2001) 2nd edition, Available in paperback from Springer or Amazon.co.uk
Recommended
R.P. Burn, Numbers and Functions, Steps into Analysis, Second Edition, Cambridge University Press (2000). Available in paperback from CUP or Amazon.co.uk
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 


