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MA1151 Introductory Real Analysis

MA1151 Introductory Real Analysis

Credits: 10 Convenor: Dr. F. Neumann Semester: 2 (weeks 15 to 26)
Prerequisites: essential: MA1101, MA1102
Assessment: Coursework and tests: 20% One and a half hour examination: 80%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Subject Knowledge

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 sequences, limits, continuity and differentiability.

Learning Outcomes

Students should know the definitions of and understand the key concepts introduced in this module, like real numbers, sequences and limits, continuity and differentiability of real functions.

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

Students should understand the difference between the real and rational numbers and be able to solve routine problems on the convergence of sequences, the existence of limits of functions and on the continuity and differentiability of functions.

Methods

Class sessions and supervision groups.

Assessment

Marked problem sheets, class tests, written examination.

Subject Skills

Aims

To provide students with problem solving skills and develop written communication skills.

Learning Outcomes

Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form.

Methods

Class sessions, supervison groups.

Assessment

Marked problem sheets, class test, written examination.

Explanation of Pre-requisites

Use is made of the following concepts from the modules MA1101 and MA1102: the notion of proof in general, concept of ordered field, rational numbers, concept of a function.

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 $\sqrt 2$ is not a rational, the equation $x^2-2=0$ has no solution in the rational numbers, equivalently, there is no rational point at which the graph of the function $y=x^2-2$ crosses the $x$-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.

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. 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.

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. Proof of the intermediate value theorem.

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

Recommended:

M. Spivak, Calculus, Benjamin Cummings.

K. G. Binmore, Mathematical Analysis, Cambridge.

R. Hoggarty, Fundamentals of Mathematical Analysis, Addison-Wesley.

S. Abbott, Understanding Analysis, Springer.

R. G. Bartle, D. R. Sherbert, Introduction to Real Analysis, Jon Wiley & Sons..

J. Lewin, An Interactive Introduction to Mathematical Analysis, Cambridge.

R. A. Adams, Calculus, Addison-Wesley.

Background:

T. M. Apostol, Mathematical Analysis, Addison-Wesley.

J. C. Burkill, A First Course in Mathematical Analysis, Cambridge.

M. Hart, An Guide to Analysis, Macmillan.

J. B. Read, An Introduction to Mathematical Analysis, Oxford.

Resources

Problem sheets, additional handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1152 Introductory Linear Algebra Up: ModuleGuide03-04 Previous: MA1102 Algebraic Structures and Number Systems

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Last updated: 2004-02-21
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