Course Title:           Introduction to Real Analysis I

Course Code:          MATH 2277

Level:                      2

Number of Credits:  3     

Semester:                1, 2

Pre-requisite(s):        MATH1141, MATH1142, MATH1151 and MATH1152 (or equivalents)

 

Course Rationale

Real analysis is a large field of mathematics based on the properties of the real numbers and the ideas of sets, functions, and limits. It is the theory behind calculus, differential equations, and probability, and it is part of the essential foundation for advanced study in many areas of pure and applied mathematics. A study of real analysis allows for an appreciation of the many interconnections between areas of mathematics.

This course aims to give students a substantial knowledge of the concepts:  limits, continuity, experience with epsilon-delta definitions and the construction of proofs. It is a core course for the mathematics major, and could be useful for students pursuing a major in physics.

 

Course Description

This is a classical course in analysis, providing a foundation for many other mathematical courses. Knowledge of Calculus, analytical geometry and basic set theory is required.

The course exposes students to rigorous mathematical definitions of limits of sequences of numbers and functions, classical results about continuity and series of numbers and their proofs. A major emphasis is placed on the proper use of definitions for the rigorous proof of theorems. The following topics will be covered: The real number system, topological properties of real numbers, sequences, continuity and differentiation.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly problem sheets, allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures, and periodic marked assignments. Assessment will be based on the marked assignments and in-course tests followed by a final examination based on the whole course

 

Learning Outcomes

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

  • Explain the completeness of a system of real numbers: a least upper bound, a greatest lower bound.
  • Elaborate on the topological concepts of the real numbers: open sets, closed sets, accumulation points, closure, open covers, compact sets.
  • Define and utilize the following concepts: sequence, subsequence, monotone sequence, Cauchy sequence.
  • Prove that a given function is continuous or discontinuous and classify its points of discontinuity.
  • Justify the convergence/divergence of a given number series;
  • Prove some of the classical theorems of real analysis.

 

Course Contents

  1. Basics: review on tautologies, contradictions and equivalence, open sentences and quantifiers. Methods of proof in Mathematics: Direct Method, Contrapositive method, Contradiction Method. Sets& Functions: Indexed families of sets and its properties, image & inverse image of functions and its properties. 
  2. Real Number Systems: Countability of ℝ and ℚ. Order properties of ℚ and its order incompleteness. Construction of ℝ from ℚ using Dedekind cuts/ axioms. Order completeness of ℝ: The least upper bound property and equivalent conditions including the nested interval property. Bounded sets, and their properties, supremum and infimum of sets. Bolzano-Weierstrass theorem.
  3. Topological Properties of ℝ: Interior points, limit points, isolated points, boundary points, frontier points, open sets, closed sets, perfect sets, compact sets and connectedness.
  4. Limits and continuity: Basic properties of continuous functions. Operations on sequences. Uniform continuity. Bounded functions. Continuous functions defined on a compact set; their boundedness, attainment of bounds, and uniform continuity. Intermediate Value Theorem. Discontinuities. Monotonic functions.
  5. Sequences: Bounded sequences, monotone sequences and their convergence, limsup and liminf and convergence criterion using them, subsequences, Cauchy sequences and their convergence criterion
  6. Series: Series of numbers, Infinite series and their convergence, Geometric series, the comparison test, Series of non-negative terms. The condensation test, Integral test (without proof). Ratio and root tests. Absolute and conditional convergence. Power Series. Alternating series and Leibnitz's theorem

Teaching Methodology

 Students will be exposed to the theoretical aspects of the mathematical analysis through informative lectures. Tutorials provided as needed during scheduled lecture hours will reinforce their learning and provide experience with the practical application of different type of problems.

The total estimated 39 contact hours may be accounted for as follows: 26 hours of lectures and 13 hours of tutorials. Course material, including practice problems, will be posted on myelearning.

 

Assessment     

The course assessment has three components:

 

Final exam:  2-hour written paper                       50%

Two Midterm Exams     (15% each)                   30%

Five Written Assignments (4% each)                  20%

 

Course Calendar

 

Week

Lecture Topics

Assignments

Tutorial

1

Course Overview/Introduction

 

Introduction: review on tautologies, contradictions and equivalence, open sentences and quantifiers. Methods of proof in Mathematics: Direct Method Contrapositive method, Contradiction Method.

None

Tutorial #1

2

Sets& Functions: Indexed families of sets and its properties, image & inverse image of functions and its properties. 

Assignment 1 given

Tutorial #2

3

Real Number Systems: Countability of ℝ and ℚ. Order properties of ℚ and its order incompleteness. Construction of ℝ from ℚ using Dedekind cuts/ axioms.

None

Tutorial #3

4

Order completeness of : The least upper bound property and equivalent conditions including the nested interval property. Bounded sets, and their properties, supremum and infimum of sets. Bolzano-Weierstrass theorem.

Assignment 1 due

Assignment 2 given

Tutorial #4

5

Topological Properties of ℝ: Interior points, limit points, isolated points, boundary points, frontier points, open sets, closed sets, perfect sets, compact sets and connectedness.

Coursework Exam #1

(15%)

No tutorial this week

6

Limits and continuity: Basic properties of continuous functions. Operations on sequences. Uniform continuity. Bounded functions

Assignment 2 due

Assignment 3 given

Tutorial #5

7

Continuity & Compact: Continuous functions defined on a compact set; their boundedness, attainment of bounds, and uniform continuity. Intermediate Value Theorem. Discontinuities. Monotonic functions.. 

None

Tutorial #6

8

Sequences: Bounded sequences, monotone sequences and their convergence, limsup and liminf and convergence criterion using them

Assignment 3 due

Assignment 4 given

Tutorial #7

9

Subsequences, Cauchy sequences and their convergence criterion and proving important theorems

None

Tutorial #8

10

Series: Series of numbers, Infinite series and their convergence, Geometric series, the comparison test,

Assignment 4 due

Tutorial #9

11

Series of non-negative terms. The condensation test, Integral test (without proof). Ratio and root tests. Absolute and conditional convergence.

Coursework Exam #2

(15%)

No Tutorial this week

12

Power Series. Alternating series and Leibnitz's theorem

None

Tutorial #10

13

Revision

None

None

 

 

Reference Material

Books:

 Prescribed

              The required text is Analysis, with an Introduction to Proof, by Steven R. Lay, Fourth          Edition, Pearson (Prentice Hall) 2005.

Recommended

              Bartle, Robert G. &Sherbert, Donald R., INTRODUCTION TO REAL ANALYSIS, third            ed., John Wiley & Sons, Inc., 2000.

              ‘Methods of Real Analysis’ by Richard R. Goldberg.-Oxford and IBH Publishing co.

LANG, Serge. Undergraduate Analysis (Undergraduate Texts in Mathematics), second edition, Springer, 2010.ISBN: 0387948414.

 

These books are pedagogically sound, comprehensively address all element of the syllabus, and provide useful case studies and examples.  

 

 Online Resources:

 The Math Forum Internet Mathematics Library.

 

This is a list of online resources for Real Analysis, including online lecture notes, software. The site is maintained by the Goodwin College of Professional Studies at Drexel University.

This free on-line textbook is a one semester course in real analysis, taught in the fall semester of 2009 at the University of Illinois at Urbana –Champaign