Math311

Course Description form

Faculties and departments requiring this course (if any): Faculty of

          Science/Department of Mathematics.

Objectives: 

The aim of this course is to introduce the students to the core of the subject

in order to extend their understanding of basic analysis and set theory. By

the end of the course, the students will be able to:

1- Know the algebraic properties and order properties.

2- Understand the completeness axiom.

3- Define convergent sequences.

4- Apply limit theorems for sequences.

5- Define monotone sequences and subsequences.

6- Recognize limit superior and inferior of a sequence.

7- Understand open and closed sets in R.

8- Recognize limit point of a set.

9- Apply the Bolzano-Weierstrass Theorem.

10- Describe the concept of compactness.

11- Understand and apply the Heine-Borel Theorem.

12- Compare the concepts of continuity and uniform continuity.

13- Evaluate the derivative.

14- Apply the Mean Value Theorem and L’Hospital’s Rule.

Contents:

The topics to be covered are roughly as follows: 

1. The Real Number System

a)    Algebraic properties

b)    Order properties

c)    Completeness axiom and its consequences

2. Sequences in of real numbers

  a) Convergent sequences

  b) Limit theorems

  c) Monotone sequences

  d) Subsequences

  e) Limit superior and inferior

  f) Cauchy sequences

3. Topology of the real line

  a) Open sets

  b) Closed sets

  c) Limit point of a set

  d) Bolzano-Weierstrass Theorem

            e) Compact sets

  f) Heine-Borel Theorem

4. Limits and continuity

          a) Limits     

         b) Continuity         

         c) Uniform continuity

4. Differentiation

         a) The derivative     

         b) Mean Value Theorem

         c) L’Hospital’s Rule

Course Outcomes:

 A- Knowledge:

The students will know the following concepts:

• the real number systems;
• sequences of real numbers;
• topology of the real line;
• limits and continuity;
• differentiation.

B-Cognitive Skills:

     The students will be expected to demonstrate their learning by using rigorous mathematical thought processes in topology of real line, differentiability on the real line, sequences of real numbers, and continuity and uniform continuity of real functions.

 C- Interpersonal skills and responsibilities:

    The students will be able to carry out independent investigation of topics of real analysis and interact with the fellow students to obtain sufficient information and then figure out solutions to problems that arise in their studies.

D- Analysis and communication:

     The students will be expected to involve in activities such as group discussion and report-writing, which play a key role in enhancing communication skills. They will work independently on homework assignments and jointly on report-writing to analyze their knowledge. 

     Assessment methods for the above elements: Through quizzes, homework assignments and exams.

Text book:

          R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis (3^rd Editon),

          John Wiley, New York, 2000.

Supplementary references

1. William R. Wade,  An introduction to Analysis (3^rd Edition) , Pearson

 Prentice Hall,  New Jersey, 2004.        

2. Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer, New York, 2003.

3. M. Stoll. Introduction to Real Analysis, 2^nd Edition, Addison-Wesley Longman, Boston, 2001.