Aims:

This module gives a rigorous introduction to some of the principles of mathematical analysis that are essential in most aspects of modern mathematics.

 Content: 

• The real numbers: Supremum and infimum, completeness axiom, rational and irrational numbers
• Sequences: Convergence, algebra of limits, Cauchy sequences, monotonicity, subsequences, Bolzano-Weierstrass Theorem
• Series: Convergence and divergence, tests, absolute convergence, rearrangements, the number e
• Continuity: Functions, formal definition of continuity, continuity and limits, algebra of continuous functions, the intermediate value theorem and the extreme value theorem

Objectives:

In addition to mastering the contents listed above, by the end of the module students will be able to understand and write formal mathematical sentences (aided by symbolic quantifiers).

 Books:

D. Stirling, Mathematical Analysis and Proof, 1997
M. Spivak, Calculus, Benjamin
M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises)
G.H. Hardy, A Course of Pure Mathematics, CUP