	Department of Mathematics & Computer Science

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MA1151 Introductory Real Analysis

MA1151(=MC146) Introductory Real Analysis

Credits: 10

Convenor: Dr Frank Neumann

Semester: 2

Prerequisites:	essential: MA1101(=MC144), MA1102(=MC145)
Assessment:	Coursework and tests: 20%	One and a half hour examination: 80%

Lectures:	18	Problem Classes:	none
Tutorials:	5	Private Study:	52
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

This module continues the discussion of number systems from MA1102; in particular, familiarity with the rational numbers will be assumed. The module will contain rigorous mathematical discussion of the central concepts and so will draw on the notions of proof and logical argument introduced in MA1101 and applied in MA1102. Familiarity with the idea of a function, as introduced in these previous modules, will also be important.

Course Description

This course will introduce students to the beginnings of Real Analysis, the study of the mathematics of the continuous number line. For a variety of reasons, it turns out that there are not enough rational numbers; for example, as $\sqrt2$ is not a rational, the equation

has no solution in the rational numbers, equivalently, there is no rational point at which the graph of the function

crosses the

-axis. To remedy this situation the real numbers are invented, but that leads to questions such as `how do you actually define the real numbers?', `how do you know when you have defined enough real numbers?' and `how do the real numbers differ from the rational numbers?' This module attempts to answer these questions.

Aims

This module aims to introduce the basic ideas of mathematical analysis and to familiarise students with the elementary properties of the real numbers and of the concepts of continuity, sequences and limits.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To understand the difference between the real and rational numbers.

To be able to solve routine problems on the continuity of functions, the convergence of sequences, the existence of limits of functions and on the differentiability of functions.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

Syllabus

Questions of the nature of the real numbers and how they differ from the rationals. Irrational numbers. The axioms of the real numbers. Definition of supremum and examples. Proofs of basic properties of suprema. Statement of the completion property of the reals. Proof of the intermediate value theorem. Infima as `dual' ideas to those concerning suprema. Construction of the real numbers.

Concept of a sequence and the notion of convergence to a limit. Examples. Proof of results on sums, products and quotients of convergent sequences. Proof of monotone convergence theorem. Applications to computations of limits of sequences defined by rational polynomials and by inductive formulæ. Second sketch construction of the real numbers.

The limit of a function; continuity via limits. Application to proving results on sums and products of continuous functions from the work on sequences. Examples (with proofs) of functions having or not having limits at certain points.

Continuity via $\delta$ , $\epsilon$ language. Proof of the equivalence between the limit definition of continuity and the $\delta$ , $\epsilon$ version. Examples, with full proof, of continuous and of non-continuous functions. Proof of continuity of sums, products and composites of continuous functions; continuity of polynomial functions.

Application of the idea of limit to rigorous definition of differentiablity. Proof of differentiablity of polynomial functions, via that of sums, products etc. Examples (with proofs) of functions differentiable or not differentiable at certain points. Proof that differentiable implies continuous.

Reading list

Details of Assessment

The final assessment of this module will consist of 20% continuous assessment and 80% from a one and a half hour examination during the Summer exam period. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.