Aims

The aim of the module is to introduce the students to the fundamental ideas of Real Analysis: limits of sequences, infinite series, limits of real functions, continuity, differentiability and the Riemann integral. The module should encourage students to think clearly and critically and to begin to be able to prove simple statements on their own.  

Learning objectives

At the end of the module students should:

• Understand the Axiom of Completeness.
• Understand the concept of a limit.
• Be able to determine whether some simple sequences and series diverge or converge.
• Understand the differences between convergence and absolute convergence.
• Be able to prove statements on their own using "epsilon" arguments.
• Understand, and be able to test, the continuity of functions of one real variable.
• Understand the derivative as a limit.
• Understand some of the properties and consequences of continuity and differentiability.
• Understand the Riemann integral and some of its properties.

Syllabus

• Discussion of the Real Numbers with least upper bound, greatest lower bound and completeness.
• Definition and properties of the limit of a sequence, including the algebra of limits.
• Bounded Monotone Convergence.
• The exponential function.
• Subsequences and the Bolzano-Weierstrass Theorem.
• Cauchy Sequences.
• Series, including absolute and conditional convergence.
• Tests for convergence: Comparison Test, Ratio Test, Condensation Test, (including a proof that the harmonic series diverges), Alternating Series Test.
• Power Series and the Radius of Convergence.
• Limit of a function at a point, both, epsilon-delta and sequential versions, and the algebra of limits.
• Continuity of a function.
• A continuous function on a closed bounded interval attains its extrema.
• Intermediate value theorem.
• Derivative as a limit, differentiability and the algebra of derivatives.
• Rolle's theorem, the Mean Value Theorem, Taylor's Theorem.
• The Riemann integral and the Fundamental Theorem of Calculus.

Recommended texts

• *** M Hart, Guide to Analysis, Palgrave (S 7 HAR).

• ** K E Hirst, Numbers, Sequences and Series, Edward Arnold (S 2.81 HIR). 

• ** J Lewin, An Interactive Introduction to Mathematical Analysis, Cambridge University Press

• ** Finney, Weir & Giordano, Thomas' Calculus, Addison-Wesley

• * T M Apostol, Calculus, Volume I, Wiley (QUARTO S 7 APO).

• * M Spivak, Calculus, Benjamin (S 7 SPI).

Teaching

• 3 lectures and 1 seminar per week in Weeks 2-10 of the Spring term and Weeks 1-3 of the Summer term
• 2 revision classes Week 4 of the Summer term.

Assessment

• Three-hour examination in Weeks 5-7 Summer term (100%)

Elective information

A first course in Mathematical Analysis, covering real numbers, limits of sequences, series, convergence, absolute convergence, power series, continuity, differentiability and the Riemann integral.

Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.

Prerequisites

• Familiarity with algebraic manipulation of equalities and inequalities, the notation of set theory, proof techniques including induction and contradiction.