xiv
TO THE STUDENTS
or by drawing parallels within the realm of mathematics and so on. Insightful
discussions and a plan of attack (called as strategy in the book) precede almost
every proof. Then we carefully explain how these ideas translate into rigoro
14
CHAPTER 1. REAL NUMBER SYSTEM
bound for S. If this is false, then there exists t S such that t > N . But, then
we have
tn > N n N > ,
a contradiction, since for any t S, we have tn . Hence we conclude that N
is an upper bound for S. Thus, S is a nonemp
24
CHAPTER 1. REAL NUMBER SYSTEM
x+2
(2) A := cfw_x R : x1
< 4 = (, 1) (2, ). The temptation would be to
clear off the fraction by multiplying both sides of the inequality by x 1. We
need to take care of the sign of x 1.
Case 1. x 1 > 0. Then we obtain
x+
1.3. LUB PROPERTY AND ITS APPLICATIONS
11
Proof. Since b a > 0, by AP2, there exists n N such that n(b a) > 1. Let
k = [na] and m := k + 1. Then clearly, na < m. We claim m < nb. Look at
Figure 1.8.
[na]
na
m
nb
[na]
na
nb
m
Figure 1.8: Density of Q.
If m
1.3. LUB PROPERTY AND ITS APPLICATIONS
9
The next couple of results are easy consequences of the Archimedean property.
Let not the simplicity of their proofs deceive you. They are perhaps the most
useful tools in analysis.
Theorem 1.3.5. (1) Given x > 0,
12
CHAPTER 1. REAL NUMBER SYSTEM
The next result generalizes the well-known fact that
2 is irrational.
Proposition 1.3.16. Let p be any prime. Then there exists no rational number
r such that r2 = p.
Proof. Let r = m/n Q be such that r2 = p. We assume tha
Statistics
Based on the authors combined 35 years of experience in teaching,
A Basic Course in Real Analysis introduces students to the aspects
of real analysis in a friendly way. The authors offer insights into the
way a typical mathematician works obser
1.4. ABSOLUTE VALUE AND TRIANGLE INEQUALITY
25
Hence x A+ iff x satisfies all the three conditions. Note that the first
condition does not involve x and is always true. Hence, any x R satisfies
the first condition. If x satisfies the second condition x >
8
CHAPTER 1. REAL NUMBER SYSTEM
LUB Property of R:
Given any nonempty subset of R which is bounded above in R, there
exists R such that = lub A.
Thus, any subset of R which has an upper bound in R has the lub in R.
Note that lub A need not be in A. See Ex
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
is an upper bound of A. . . . . . .
is not an upper bound of A. . . . .
Upper bound of a set is not unique.
Set which is not bounded above. . .
is the
6
CHAPTER 1. REAL NUMBER SYSTEM
Proposition 1.2.5. Let A R be a nonempty subset bounded above in R. If
and are least upper bounds of A, then = , that is, the least upper bound of
a nonempty subset bounded above in R is unique.
Proof. Since is a least upp
To the Students
We wrote this book with the aim of this being used for self-study.
There are many excellent books in real analysis. Many of them are student
friendly. Their exposition is clear, precise, and exemplary more often than not. All
of them, if w
xii
PREFACE
-, -n0 treatment (which may be considered as quantitative). Students who are
exposed very early to the topological notions are invariably very uncomfortable
while dealing with some proofs where the arguments fall under the name hardanalysis.
A
16
CHAPTER 1. REAL NUMBER SYSTEM
We observe that a < x < z < y b. Hence it follows that a < z < b and hence
a < z b.
Note that we have used and as a place holder in the definitions of
the last five subsets. In the definition of (a, ), there is one and onl
1.3. LUB PROPERTY AND ITS APPLICATIONS
19
(10) Let A, B R be nonempty. Define
A + B := cfw_x R : (a A, b B) such that x = a + b
= cfw_a + b : a A, b B.
(a) Let A = [1, 2] = B. What is A + B?
(b) Let A = B = N. What is A + B?
(c) Let A = B = Z. What is A +
Preface
This book is based on more than two decades of teaching Real Analysis in the
famous Mathematics Training and Talent Search Programme across the country
in India.
The unique features of our book are as follows:
1) We create an interest in Analysis
1.3. LUB PROPERTY AND ITS APPLICATIONS
7
Remark 1.2.7. Most often the proposition above is used by taking =
for some > 0.
Example 1.2.8. If an upper bound of A belongs to A, then lub A = . Thus
the maximum of a set, if it exists, is the LUB of the set.
E
1.2. UPPER AND LOWER BOUNDS
5
A
x
x
Figure 1.4: Set which is not bounded above.
Can you visualize this in a number line?
When do you say A R is not bounded below in R?
An upper bound of a set need not be an element of the set. For example, if
A = (a, b) i
20
CHAPTER 1. REAL NUMBER SYSTEM
1.4
Absolute Value and Triangle Inequality
Definition 1.4.1 (Absolute value of a real number). For x R, we define
(
x, if x > 0
|x| =
x, if x 0.
Note that |x| = maxcfw_x, x. If we draw the graphs of the functions f : x 7 x
1.3. LUB PROPERTY AND ITS APPLICATIONS
13
We show that cn < or cn > cannot happen. See Figure 1.10. If cn < ,
the picture shows that we can find c1 > c, c1 very near to c, such that
we still have cn
1 < , c1 S. This shows that c1 S. Hence c1 c, a
contradi
Chapter 1
Real Number System
Contents
1.1
1.2
1.3
1.4
Algebra of the Real Number System . .
Upper and Lower Bounds . . . . . . . .
LUB Property and Its Applications . .
Absolute Value and Triangle Inequality
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
10
CHAPTER 1. REAL NUMBER SYSTEM
Proof. Let S := cfw_k Z : k x. We claim that S 6= . For, otherwise, for each
k Z, we must have k > x. Let n N be arbitrary. Then k = n Z and
hence n = k > x. It follows that n < x. Hence x is an upper bound for N,
contradi
22
CHAPTER 1. REAL NUMBER SYSTEM
to the right half the distance from the midpoint, then we must get the maximum.
That is, maxcfw_a, b = (a + b)/2 + |a b| /2.
Having guessed this, it is easy to verify it. Let b > a. Then maxcfw_, b = b. and
hence |a b| /2
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
2014 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Versi
Contents
Preface
xi
To the Students
xiii
About the Authors
xv
List of Figures
xvii
1 Real Number System
1.1 Algebra of the Real Number System . .
1.2 Upper and Lower Bounds . . . . . . . .
1.3 LUB Property and Its Applications . . .
1.4 Absolute Value and
TWISTED STABLE MAPS TO TAME ARTIN STACKS
arXiv:0801.3040v1 [math.AG] 19 Jan 2008
DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI
Contents 1. Introduction 2. Twisted curves 3. Interlude: Relative moduli spaces 4. Twisted stable maps 5. Reduction of Space
2
CHAPTER 1. REAL NUMBER SYSTEM
(A4) For every x R, there exists a unique element y R such that x + y = 0.
We denote this y by x. (Existence of Additive Inverse)
The properties of the multiplication of real numbers are listed below.
(M1) x y = y x for all
4
CHAPTER 1. REAL NUMBER SYSTEM
A
x
Figure 1.2: is not an upper bound of A.
If is an upper bound of A and 0 > , then 0 is also an upper bound of A.
See Figure 1.3.
A
0
Figure 1.3: Upper bound of a set is not unique.
Lower bounds of a nonempty subset of R
About the Authors
Dr. Ajit Kumar is a faculty member at the Institute of Chemical Technology,
Mumbai, India. His main interests are differential geometry, optimization and the
use of technology in teaching mathematics. He received his Ph.D. from Universit
1.3. LUB PROPERTY AND ITS APPLICATIONS
15
How to get the algebraic identity
(xn y n ) (x y) [xn1 + xn2 y + + xy n2 + y n1 ]?
Recall how to sum a geometric series: sn := 1 + t + t2 + + tn1 . Multiply
both sides by t to get tsn = t + + tn . Subtract one fro