Components of Assessment	Method(s)	Percentage Weighting
Coursework	Two assignments worth 30% each	60%
Class Test	One class test	40%

None.

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, or a function to be continuous. Historic motivation and the rigorous use of definitions and logic play a central role. Furthermore, tools such as convergence tests are presented and their validity proved.

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.
• prove or disprove continuity of real-valued functions based on the definition and basic properties of continuity.
• argue logically to justify and follow simple proofs and counterexamples to statements involving real-valued functions.
• quote and apply basic theorems in analysis, notably the Theorem of Bolzano-Weierstrass and convergence tests.

• The axioms of real numbers. Denseness of rational and irrational numbers. The least upper bound property and consequence to existence of limits. The triangle inequality.
• Concept of proofs, proof by natural induction, set theoretic notation and quantifiers.
• Sequences and convergence, and their properties. Boundedness, subsequences and the Theorem of Bolzano-Weierstrass.
• Infinite series, convergence and absolute convergence. Convergence tests. Power series and radius of convergence. Rearrangement of series.
• Historical motivation of analysis in mathematics.

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

3 lecture/tutorial hours per week for 12 weeks.

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). 

S. Lang, Analysis I Addison-Wesley (1968).

16th July 2007