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MTH241Sciences3 Unitsintermediate

Introduction To Real Analysis

This course introduces students to the fundamental concepts of real analysis. It covers preliminaries such as sets, functions, and mathematical induction. The course delves into the properties of real numbers, including algebraic, order, and completeness properties. Students will also explore sequences, series, limits, continuity, differentiation, and their associated theorems, providing a solid foundation for advanced mathematical studies.

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156h
Study Time
13
Weeks
12h
Per Week
advanced
Math Level
Course Keywords
Real AnalysisSequencesLimitsContinuityDifferentiation

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
∑
Math Level
Advanced Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Set Theory

2

Mathematical Induction

3

Real Number System

4

Sequences

5

Limits

6

Continuity

7

Differentiation

Total Topics7 topics

Requirements

Knowledge and skills recommended for success

MTH111

MTH121

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assessments

Comprehensive evaluation of course material understanding

Written Assessment

final examination

Comprehensive evaluation of course material understanding

Computer Based Test

Career Opportunities

Explore the career paths this course opens up for you

Data Analyst

Apply your skills in this growing field

Financial Analyst

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Research Scientist

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Statistician

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Actuary

Apply your skills in this growing field

Industry Applications

Real-world sectors where you can apply your knowledge

FinanceData ScienceEngineeringResearchAcademia

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Preliminaries

4h

Unit 1: Sets and Functions

4 study hours
  • •Review set theory notations and definitions.
  • •Practice identifying subsets, unions, intersections, and complements.
  • •Work through examples of sets defined by properties.
Week
2

Module 1: Preliminaries

4h

Unit 2: Mathematical Induction

4 study hours
  • •Study the principle of mathematical induction.
  • •Practice proving statements using mathematical induction.
  • •Understand the well-ordering property of natural numbers.
Week
3

Module 1: Preliminaries

4h

Unit 3: Finite and Infinite Sets

4 study hours
  • •Learn the definitions of finite, infinite, denumerable, countable, and uncountable sets.
  • •Understand and apply the uniqueness theorem.
  • •Practice identifying countable and uncountable sets.
Week
4

Module 2: The Real Numbers

4h

Unit 1: The Algebraic and Order Properties of R

4 study hours
  • •Study the algebraic properties of real numbers.
  • •Understand the order properties and trichotomy property.
  • •Practice working with inequalities and proving basic theorems.
Week
5

Module 2: The Real Numbers

4h

Unit 2: Absolute Value and Real Line

4 study hours
  • •Understand the definition and properties of absolute value.
  • •Apply the triangle inequality and its variations.
  • •Practice solving inequalities involving absolute values.
Week
6

Module 2: The Real Numbers

4h

Unit 3: The Completeness Property of R

4 study hours
  • •Study the completeness property of real numbers.
  • •Understand suprema and infima.
  • •Apply the completeness property to prove fundamental results.
Week
7

Module 2: The Real Numbers

4h

Unit 4: Applications of the Supremum Property

4 study hours
  • •Apply the Archimedean property.
  • •Understand the existence of square roots.
  • •Study the density of rational numbers in R.
Week
8

Module 2: The Real Numbers

4h

Unit 5: Intervals

4 study hours
  • •Understand the nested interval property.
  • •Study binary and decimal representations of real numbers.
  • •Prove the uncountability of R.
Week
9

Module 3: Sequences and Series

4h

Unit 1: Sequences and Their Limits

4 study hours
  • •Learn the definition of a sequence and its limit.
  • •Apply the uniqueness of limits theorem.
  • •Practice proving convergence using the definition of limit.
Week
10

Module 3: Sequences and Series

4h

Unit 2: Limit Theorems

4 study hours
  • •Study the limit theorems for sequences.
  • •Apply the squeeze theorem.
  • •Understand the properties of bounded sequences.
Week
11

Module 3: Sequences and Series

4h

Unit 3: Monotone Sequences

4 study hours
  • •Understand monotone increasing and decreasing sequences.
  • •Apply the monotone convergence theorem.
  • •Study Euler's number.
Week
12

Module 3: Sequences and Series

4h

Unit 4: Subsequences and the Bolzano-Weierstrass Theorem

4 study hours
  • •Learn the definition of a subsequence.
  • •Understand the Bolzano-Weierstrass theorem.
  • •Apply divergence criteria.
Week
13

Module 3: Sequences and Series

4h

Unit 5: The Cauchy Criterion

4 study hours
  • •Understand the Cauchy criterion for convergence.
  • •Apply the Cauchy criterion to determine convergence.
  • •Study contractive sequences.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Thoroughly review all definitions and theorems from each unit.

2

Focus on understanding the proofs of key theorems.

3

Practice solving a variety of problems from the textbook and assignments.

4

Create concept maps linking different units and modules.

5

Allocate sufficient time for reviewing and practicing problems.

6

Prioritize understanding over memorization.

7

Form study groups to discuss challenging concepts and solutions.

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