Introductory analysis
4 points * First semester * Clayton * Prerequisites: a 12-point sequence in first year mathematics including MAT1010 or MAT1020 (a credit grade or higher in MAT1050 is an acceptable alternative to a pass in MAT1010) * Prohibitions: GAS2611, MAP2011
Objectives On the completion of this subject student will understand basic concepts in several areas of mathematics which defy intuition, especially those that involve infinite processes, such as dealing with the notion of the limit of a sequence; appreciate the importance of proof and the use of rigorous arguments; be acquainted with the elementary topology and algebra of the real line; have a skill of determining the convergence or otherwise of sequences and series; be able to use power series and Fourier series; understand abstract notions such as limits, convergence of sequences and series and continuity; appreciate the use of analytical methods in many areas of pure and applied mathematics.
Synopsis Basic properties of numbers, inequalities, functions and graphs, limits of functions, continuity, least upper bounds, immediate value theorem, applications to solvability of polynomial equations, mean number theorem and fundamental theorem of calculus; sequences of real numbers, subsequences, convergence theorems, Cauchy sequences, applications including contraction mapping theorem, iterative methods for finding roots and fixed points, Newton's method; series of real numbers, test for convergence, methods of summation; sequences of functions, uniform convergence, uniform continuity, interchange of limit theorems, series of functions, power series, Fourier series.
Assessment Examination (1.5 hours): 85% * Tests and/or assignments: 15%
Prescribed texts
Spivak M Calculus 2nd or 3rd edn, Publish or Perish
Recommended texts
Krantz S G Real analysis and foundations, studies in advanced mathematics CRC, 1992
Brabenec R L Introduction to real analysis, PWS-Kent, 1990
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