Lecture Notes in Elementary Real Analysis

By Rohan Dalpatadu

Elementary Real Analysis is a vital component of every Bachelors degree in Mathematics and Statistics. This book provides a somewhat detailed introduction to the subject. It may be used in an Introductory Real Analysis course as a main text or reference.

• Language: English
• Publisher: Trafford Publishing
• Release date: Oct 13, 2015
• ISBN: 9781490764719

Rohan Dalpatadu

Dr. Rohan J. Dalpatadu is Associate Professor of Mathematics in the Department of Mathematical Sciences at University of Nevada, Las Vegas (UNLV). He earned his B.Sc. (Honors) (1974) in Mathematics from the University of Ceylon, Colombo, M.S. (1981) and Ph.D. (1986) in Mathematics from Southern Illinois University, Carbondale. He is also an Associate (1991) of the Society of Actuaries. He has taught undergraduate and graduate mathematics, statistics, and actuarial science courses. His research interests include numerical analysis, applied mathematics, applications of probability and statistics in gaming and actuarial science. His email address is [e-mail].

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Lecture Notes in Elementary Real Analysis

Rohan Dalpatadu

© Copyright 2015 Rohan Dalpatadu.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author.

Print information available on the last page.

isbn: 978-1-4907-6472-6 (sc)

isbn: 978-1-4907-6471-9 (e)

Library of Congress Control Number: 2015915458

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Contents

Preface

Chapter 1 The Real Number System

1.1   Sets and Operations on Sets

1.2   Relations and Functions

1.3   Mathematical Induction

1.4   The Real Number System

1.5   Countable and Uncountable Sets

Chapter 2 Sequences and Series of Real Numbers

2.1   Convergent Sequences

2.2   Limit Theorems

2.3   Monotonic Sequences

2.4   Subsequences and the Bolzano-Weierstrass Theorem

2.5   Limit Superior and Limit Inferior

2.6   Cauchy Sequences

2.7   Series of Real Constants

2.8   Further Tests for Convergence and Absolute Convergence

Chapter 3 Limits and Continuity

3.1   Structure of Point Sets in 

3.2   Limit of a Function

3.3   Continuous Functions

3.4   Uniform Continuity

3.5   Discontinuities and Monotonic Functions

Chapter 4 Differentiation

4.1   The Derivative

4.2   Mean Value Theorems

Chapter 5 Integration

5.1   The Riemann Integral

5.2   Properties of the Riemann Integral

5.3   Fundamental Theorem of Calculus

5.4   Improper Riemann Integrals

Chapter 6 Sequences and Series of Functions

6.1   Sequence of Functions

6.2   Series of Functions

6.3   Power Series and Special Functions

Bibliography

PREFACE

These lecture notes are the ones presented to my Elementary Real Analysis classes in the fall semester of 2014 and the spring semester of 2015 and are solely based on two well written texts on the subject: Principles of Mathematical Analysis by Walter Rudin, 3rd Edition, McGraw-Hill, Inc. 1976 and Introduction to Real Analysis by Manfred Stoll, Addison Wesley Longman, Inc. 2001. The majority of examples and assignments were also from these two texts.

The first three chapters were covered in the fall semester and the last three chapters in the spring semester. Due to the expanded nature of the notes, the material covered in the two semesters appear to be somewhat less than that of typical courses across the nation. However, this enabled most of the students to obtain a deeper understanding of the subject and also the techniques used in the proofs.

I have only assumed that the students have been introduced to the rational numbers and their properties and developed the real number system based on this understanding. The assignments and the examples used just the functions that have been introduced in the course, e.g., the exponential function was only used in the examples after it had been defined along with its properties. The trigonometric functions and hyperbolic functions were defined almost towards the end of the course. However, anyone wishing to use these notes can always include problems and examples based on functions to be introduced later on.

Chapter 1 introduces the Real Number System with a Completenes Axiom; I have deliberately avoided Dedekind cuts to make the completeness more understandable to undergraduate students. Furthermore, the Principle of Mathematical Induction was also presented because of its varied applications. Chapter 2 concentrates on the development of sequence and series of constants and explains in detail almost all of the essential results on them. Chapter 3 is on limits and continuity of real valued functions and I have briefly introduced Point Set Topology in order to present elegant proofs of some of the theorems. Chapter 4 discusse the basic ideas of differentiation with some Mean Value Theorems. In Chapter 5 on integration, I have deliberately avoided the Riemann Stieltjes Integral because the results and proofs are quite similar and also because this concept can be handled quite easily, if the students have a solid understanding of the basic ideas. Chapter 6 is where, the most important sections of Elementary Real Analysis are covered, i.e., Sequences and Series of Functions. Here, we are able to introduce the Transcendental Functions with their properties. However, due to the expanded nature of the course, I was not able to proceed any further than this.

The lecture notes provided here should give most students a solid background in Elementary Real Analysis in order for them to be able to complete a course in (graduate level) Real Analysis. As a last note I would like to add that the text by Walter Rudin is ideal for an introductory course in (real) analysis for most advanced undergraduate mathematics majors, whereas Manfred Stoll’s text would be quite useful to almost all students. I have tried to go one step further by providing more details and examples and presenting the material without assuming results to be presented later on.

Chapter

1

The Real Number System

1.1   Sets and Operations on Sets

In this section, we shall define sets and operations on sets that the reader is already familiar with and will use standard notation. Examples are deemed not necessary for this section and have not been provided.

1.1D1   Definition:   Sets

A set is a well defined collection of objects and can be described by listing its objects or elements and also may be defined as the collection of all elements x satisfying a given property. Thus the notation

66886.png

defines A to be the set of all elements x satisfying the condition 66877.png This reads as

"A equals the set of all elements x such that 66866.png "

Notation:    66856.png means that x is an element of the set A. 66847.png means that x is not an element of the set A.

We may also use the notation 66838.png to mean that only the elements x from the set X are being used.

We will assume that the set of natural numbers or positive integers have already being defined. This set will be denoted by 66828.png .

We will also assume that the set of integers have already being defined. This set will be denoted by 66818.png .

Furthermore, the set of rational numbers with the binary operations of addition and multiplication will be denoted by 66810.png We shall also assume all the properties of 66803.png

The set without any members is called the empty (or null) set and is denoted by 66793.png

1.1D2   Definition:   Equality of Sets

Two sets A and B are said to be equal if every element of A is an element of B and every element of B is an element of A. In this case, we write 66784.png means that the set A is not equal to the set B.

1.1D3   Definition:   Subsets of Sets

A set A is said to be a subset of a set B or A is said to be contained in B if every element of A is also an element of B. In this case, we use the notation: 66775.png . If 66764.png , then we say that A is a proper subset of B and use the notation 66754.png . Note that the empty set 66745.png is a subset of any set A.

Note:    66737.png .

1.1D4   Definition:   Union, Intersection, and Complement of Sets

The union of sets A and B is the set of all elements that belong to A or B or both A and B and is denoted by 66727.png Thus,

66718.png

The intersection of sets A and B is the set of all elements that belong to both A and B and is denoted by 66708.png Thus,

66700.png

The relative complement of A in B or the complement of A relative to B, denoted by 66691.png is the set of all elements in B that are not elements of A. Thus

66683.png

If A is a subset of some fixed set X consisting of all the elements under consideration, then 66674.png is referred to as the complement of A and is denoted by 66663.png . Thus,

66653.png

1.1T1   Theorem:   Distributive and DeMorgan’s Laws

Let A, B, and C be sets. Then

(a) 66642.png .

(b) 66634.png

(c) 66625.png .

(d) 66615.png .

(a) and (b) are referred to as the distributive laws, whereas (c) and (d) are known as DeMorgan Laws.

Proof:   We will provide the proofs of (a) and (c). The proofs of (b) and (d) are left as an exercise.

(a)    66604.png

66595.png

This completes the proof of (a).

(c)    66587.png

66579.png66569.png66556.png

This completes the proof of (c).

1.1D5   Definition:   The Power Set

Let A be a set. The set consisting of all the subsets of A is called the power set of A and is denoted by 66544.png . If a set has n elements, then its power set consists of 66535.png elements, i.e., the set has 66527.png subsets.

1.1D6   Definition:   Cartesian Product

Let A and B be sets. Then the Cartesian product of A and B, denoted by 66518.png is the set of all ordered pairs 66509.png , where the first component a is an element of A and the second component b is an element of B. Thus,

66498.png

1.2   Relations and Functions

1.2D1   Definition:   Relations and Functions

Let A and B be sets. Then a subset of the Cartesian product 66489.png is called a relation from A into B. If 66481.png is a relation from A into B, then the domain of 66472.png denoted by 66463.png is given by

66451.png

The range of 66441.png denoted by 66432.png , is given by

66424.png

A subset F of the Cartesian product 66415.png is called a function from A into B if the following condition is also satisfied:

66405.png

Notation:   The domain and the range of F have already been defined because a function is also a relation. If F is a function and 66395.png then we may use the notation 66386.png and may refer to the function as 66378.png If f is a function from A to (into) B, we say that f maps A to (into) B and use the notation: 66370.png

1.2D2   Definition:   One-to-one and Onto Functions

Let 66361.png The function f is said to be onto if 66350.png In this case, we say that f is a function (mapping) from A onto B. The function f is said to be one-to-one 66339.png if the following condition is satisfied:

66330.png

Note:   The function f is one-to-one if 66321.png

1.2D3   Definition:   Image and Inverse Image

Let 66310.png

(a)   The image of E under f, denoted 66300.png , is given by

(b)   The inverse image of H under f, denoted 66289.png , is given by

66281.png

1.2D4   Definition:   The Inverse Function

Let 66273.png Then the inverse function of f, denoted by 66265.png is the function: 66255.png

The domain of 66244.png is the range of f and the range of 66234.png is