Here is the official site from the department.

Time and place: Tue 17-19 in L3; Thu 16-17 in MS.02.

Assessment: Material from this half of the course is examined in the first week of Term 2, probably Monday.

Description:

  • Inequalities: rules for manipulating inequalities; inequalities and powers; inequalities and absolute values; Bernoulli's inequality; triangle inequality.

  • Sequences: monotonic sequences; bounded sequences; subsequences; tending to infinity; null sequences (and algebra of); convergent sequences (and algebra of, boundedness, uniqueness of limit); sandwich theorems; shift rules; standard limits; ratio test; limits and inequalities; recursively-defined sequences.

  • The real numbers: infinitely many rationals/irrationals in any open interval; rationals/irrationals and terminating/recurring/non-recurring decimals; numbers with more than one decimal representation; sets and upper/lower bounds; supremum and infimum; completeness axiom (in the form "every non-empty set bounded above has a supremum"); consequences of completeness; existence of k-th roots; Bolzano-Weierstrass theorem; Cauchy sequences (contracting sequences as example).

  • Series: partial sums; convergence and divergence; sum rule; shift rule; null sequence test.

    • Series with positive terms: boundedness condition, comparison tests, ratio test, integral test.
    • Alternating series: alternating series test, absolute convergence, absolute value form of ratio test, conditional convergence, rearrangements.

Module material:

Here are lecture notes.
I occasionally put comments on my facebook page; but anything of importance can be found here.

References:

  • Mary Hart, Guide to Analysis, Macmillan Mathematical Guides, 2001, ISBN: 0333794494.

  • G. H. Hardy, A Course of Pure Mathematics, Cambridge Mathematical Library, 1993, ISBN: 0521092272.

  • Michael Spivak, Calculus, Publish Or Perish, 1994, ISBN: 0914098896.

  • David S. G. Stirling, Mathematics Analysis and Proof, Horwood Publishing, 1997, ISBN: 1898563365.