MATH201 Real Analysis

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

Analysis is, broadly, the part of mathematics which deals with limiting processes. 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. Real analysis specialises to real-valued functions of a real variable. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis. At the end of the semester, students should have a grounding in the methods of analysis which will prove invaluable in later years.

Paper title Real Analysis MATH201 Mathematics 0.15 18 points Semester 1 (On campus) \$929.55 Tuition Fees for international students are elsewhere on this website.
Prerequisite
MATH 140 or MATH 170
Restriction
MATH 353
Schedule C
Arts and Music, Science
Eligibility
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.
Contact
maths@otago.ac.nz
Teaching staff

Paper Structure

This paper will begin by developing an axiomatic description of the real line. We will then use this one-dimensional construction to develop the n dimensional Euclidean spaces and understand their properties. The bulk of this course will be taken up with understanding rigorous definitions of limits in a variety of settings.

Main topics:

• A review of the real number system
• The completeness axiom
• The Euclidean spaces
• The distance function and open and closed sets
• Limits of sequences and the algebra of limits
• Limits of series and the algebra of limits
• Continuous functions
• Limits of functions and the algebra of limits
• Applications of real analysis in one dimensional calculus
Textbooks
Textbooks are not required for this paper.
Course outline
View course outline for MATH 201
Critical thinking.
Learning Outcomes

Students who successfully complete this paper are expected to

• Understand the formal definition of Euclidean spaces, particularly the real number line and plane
• Understand the notion of open and closed sets in the Euclidean setting
• Understand the rigorous definition of convergence for a sequence or series and apply appropriate tools determine whether example sequences/series are convergent
• Understand the definition of a continuous function and the key properties of such functions
• Understand convergence of functions both pointwise and uniform

Timetable

Semester 1

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

Lecture

Stream Days Times Weeks
Attend
A1 Monday 12:00-12:50 9-15, 18-22
Wednesday 12:00-12:50 9-15, 17-22
Friday 09:00-09:50 9-14, 17-22

Tutorial

Stream Days Times Weeks
Attend one stream from
A1 Thursday 11:00-11:50 10-15, 17-22
A2 Thursday 14:00-14:50 10-15, 17-22
A3 Wednesday 14:00-14:50 10-15, 17-22

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

Analysis is, broadly, the part of mathematics which deals with limiting processes. 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. Real analysis specialises to real-valued functions of a real variable. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis. At the end of the semester, students should have a grounding in the methods of analysis which will prove invaluable in later years.

Paper title Real Analysis MATH201 Mathematics 0.15 18 points Semester 2 (On campus) Tuition Fees for 2023 have not yet been set Tuition Fees for international students are elsewhere on this website.
Prerequisite
MATH 140 or MATH 170
Restriction
MATH 353
Schedule C
Arts and Music, Science
Eligibility
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.
Contact
maths@otago.ac.nz
Teaching staff

Paper Structure

This paper will begin by developing an axiomatic description of the real line. We will then use this one-dimensional construction to develop the n dimensional Euclidean spaces and understand their properties. The bulk of this course will be taken up with understanding rigorous definitions of limits in a variety of settings.

Main topics:

• A review of the real number system
• The completeness axiom
• The Euclidean spaces
• The distance function and open and closed sets
• Limits of sequences and the algebra of limits
• Limits of series and the algebra of limits
• Continuous functions
• Limits of functions and the algebra of limits
• Applications of real analysis in one dimensional calculus
Textbooks
Textbooks are not required for this paper.
Critical thinking.
Learning Outcomes

Students who successfully complete this paper are expected to:

• Understand the formal definition of Euclidean spaces, particularly the real number line and plane
• Understand the notion of open and closed sets in the Euclidean setting
• Understand the rigorous definition of convergence for a sequence or series and apply appropriate tools determine whether example sequences/series are convergent
• Understand the definition of a continuous function and the key properties of such functions
• Understand convergence of functions both pointwise and uniform

Timetable

Semester 2

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

Lecture

Stream Days Times Weeks
Attend
A1 Monday 12:00-12:50 28-34, 36-41
Wednesday 12:00-12:50 28-34, 36-41
Friday 09:00-09:50 29-34, 36-41

Tutorial

Stream Days Times Weeks
Attend one stream from
A1 Thursday 11:00-11:50 28-34, 36-41
A2 Thursday 14:00-14:50 28-34, 36-41