## MATH201 Real Analysis

First Semester |

MATH 201 is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus.

### Paper details

Analysis is, broadly, the part of mathematics which deals with limits. Real analysis is concerned with the analysis of the real numbers, sequences and series on the real line, as well as functions of a real variable. The methods of real analysis have been developed over the past two centuries to give a rigorous theoretical foundation for limiting process such as integration and differentiation. In particular, the main examples students have met in school and first-year university are from calculus, where the derivative and integral are defined using quite different limiting processes. This paper introduces the basic ideas of real analysis. At the end of the semester, students should have a solid grounding in the methods of real analysis which will prove invaluable in later study.

### Potential students

MATH 201 is compulsory for the Mathematics major and is of particular relevance also for students majoring in Statistics, Physics or any discipline requiring a quantitative analysis of systems and how they change with space and time. It is a prerequisite for MATH 301 (Hilbert spaces) and MATH 302 (Complex Analysis).

### Prerequisites

MATH140 or MATH 170

### Course Outline

The paper will cover the following topics:

- A review of the real number system
- The completeness axiom
- The distance function and open and closed sets
- Limits of sequences and series
- The algebra of limits
- Continuous functions
- Limits of functions and the algebra of limits
- Applications of real analysis in one dimensional calculus

### Lecturer

Dr. Timothy Candy, room 216 Science III, ext 7781, tcandy@maths.otago.ac.nz

### Lectures

3 per week: Mondays at 12 noon, Wednesdays at 12 noon and Fridays at 9 am.

### Tutorial

Thursday 11am and 2pm. Tutorials start in the second week of semester.

### Assessment

The internal assessment is made up of 60% from 3 assignments and 40% from an in-class tests.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, (10E + 3A + 2T)/15)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Test mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.