Fundamentals of Real Analysis

Course aim

To introduce the student to fundamental concepts of analysis that underpin a large number of areas of mathematics, both pure and applied. By studying the material in this course the student will learn theoretical principles and techniques of analysis and learn to apply them so as to analyse and solve mathematical problems.

Course content

A selection drawn from the following topics:

Types of proof used in analysis: direct proof, proof of a contrapositive, proof by induction, proof by contradiction. Quantifiers. Negation of a statement.

Properties of the real numbers, least upper bounds, the Axiom of Completeness and its consequences.

Cardinality of a set. One-to-one and onto functions. Finite, countable and uncountable sets.

Properties of sequences and series. Limit theorems, Monotone Convergence Theorem, Bolzano-Weierstrass Theorem, Cauchy Criterion.

Subsets of the real numbers. Open sets, closed sets, compact sets, perfect sets, connected sets. Closure of a set, unions, intersections and complements.

Properties of continuous functions and of differentiable functions. Intermediate Value Theorem, Mean Value Theorem.

Possible additional topics such as Taylor series, power series, uniform convergence, Riemann integration, metric spaces.

Textbook(s)

Abbott, Stephen 2015, Understanding Analysis, Second Edition

Prerequisite(s)

Corequisite(s)

Nil

Teaching method

Note: These components may or may not be scheduled in every study period. Please refer to the timetable for further details.

Assessment

Examination, Problem solving exercise

Fees

EFTSL*: 0.125
Commonwealth Supported program (Band 1)
To determine the fee for this course as part of a Commonwealth Supported program, go to:
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Fee-paying program for domestic and international students
International students and students undertaking this course as part of a postgraduate fee paying program must refer to the relevant program home page to determine the cost for undertaking this course.

Non-award enrolment
Non-award tuition fees are set by the university. To determine the cost of this course, go to:
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Not all courses are available on all of the above bases, and students must check to ensure that they are permitted to enrol in a particular course.

* Equivalent Full Time Study Load. Please note: all EFTSL values are published and calculated at ten decimal places. Values are displayed to three decimal places for ease of interpretation.