Introduction to Real Numbers: Properties and Analysis, Lecture notes of Mathematics

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Badji Mokhtar University-Annaba Faculty of Technology Common Core Computer Engineering-1st Year 2024-

Analysis 1 course by Dr. Salah Derradji Lylia

Chapitre 1

Some Properties of the set of real

numbers

1.1 Introduction-Notable subsets of R

IThe set of natural numbers, also known as positive integers, denote by N , is deÖned as N = f 0 ; 1 ; 2 ; 3 ; :::g : I The set of integers denoted by Z, is deÖned as Z = f:::; 2 ; 1 ; 0 ; 1 ; 2 ; :::g : IThe set of decimal numbers, denoted by D, is the set of numbers of the form a = (^10) p ; p 2 N; 2 Z: For example : 1 :25 = 125102 2 D: IThe set of rational numbers, denoted by Q, is deÖned as Q =

n (^) p q ; p^2 Z; q^2 N

o

. It can be shown that any rational number pq can be expressed not only in its fractional form but also as a limited form but also as a limited decimal expansion of the form pq = 0 ; 1 2 ::: (^) n, or as an inÖnite periodic expansion. For example, for r = 35 , we have r = 0: 6 (limited decimal expansion), for r = 1= 3 , we have r = 0: 333 :::(inÖnite periodic expansion) IA number is termed irrational if it is not rational, i.e, if its decimal expansion is inÖnite and non-repeating. Examples include p2 = 1: 4142 :::;  = 3: 14159 :::; e = 2: 718 ::: Remark1. To prove that a number is irrational, we assume that is rational and we derive a contradiction. The set of real numbers, denoted by R , is the set of both rational numbers Q and irrational

1.3 Archimedes property and integer part

1.3.1 Archimedes property

Property 2 : R is Archimedean. This means that it satisÖes one of the following equivalent properties : 1-Given two real numbers y and x, where x > 0 , there exists a positive integer n 2 N^ such that y  nx (By taking enough steps of length x, we exceed y). 2-Given a real number y; there exists an integer n 2 N such that y < n.

1.3.2 Integer part

DeÖnition 1(Integer part) For any real number x, there exists a unique integer 2 Z such that :  x < + 1. The integer is called the integer part of x and is denoted as E(x) or [x]. Therefore, we have :

E(x)  x < E(x) + 1 Remark 5.The integer part of a real number x is the largest integer that is less than or equal to x. Example 1 E(2:852) = 2 because 2  2 : 852 < 3 : E( 3 :5) = 4 because - 4  3 : 5 < 3.

1.3.3 Integer part properties

1. 8 x 2 R : x 1 < E(x)  x:
2. 8 x 2 Z : E(x) = x:
3. The integer part is an increasing application :

8 x; y 2 R : x  y ) E(x)  E(y)

4. 8 x; y 2 R : E(x) + E(y)  E(x + y):
5. 8 x 2 R; 8 n 2 Z : E(x + n) = E(x) + n:

1.4 The absolute value

DeÖnition 2 Let x 2 R:The absolute value jxj of x is deÖned by

jxj =

x if x  0 x if x < 0 Properties

1. jxj  0 ; jxj = jxj ; jxj  x:
2. jx:yj = jxj jyj ; xy = jjxyjj ; y 6 = 0:
3. jxnj = jxjn^ :
4. jxj  a , a  x  a:
5. jxj  a , x  a or x  a:
6. jx  yj  jxj + jyj :(Triangle inequality)
7. jjxj jyjj  jx yj :
8. jxj = jyj , x = y or x = y:

1.5 Density of Q in R

Theorem 1 Between any two real numbers, there is a rational number, i.e.

8 x; y 2 R; x < y ) 9r 2 Q : x < r < y:

Proof Let x and y be two real numbers such that x < y; we denote z = y x > 0 : Since R is Archimedean, 9 n 2 N^ such that 1 < nz ) z > (^) n^1 : We have E(nx)  nx < E(nx) + 1

lower bound, it is said to be bounded below. If A has both an upper and a lower bound, it is said to be bounded. Example 3 : I Let A be the interval : A = [0; 1] :We have : maxA = 1 because : 1 2 A and 8 x 2 A : x  1 : minA = 0 because : 0 2 A and 8 x 2 A : x  0 : I Let B be the interval : B = ]0; 1[ :We have : 8 x 2 B : x  1 but 1 2 = B; so max B doesnít exist. We have : 8 x 2 B : x  0 but 0 2 = B; so min B doesnít exist.

1.7 Supremum and inÖmum. Completeness axiom

DeÖnition 5 If A is bounded above and the set of its upper bounds has a smallest element , then is called the least upper bound or supremum of A, denoted as = supA: If A is bounded below and the set of its lower bounds has a largest element , then is called the greatest lower bound or inÖmum of A, denoted as = inf A. Example 4 1- A 1 = ]0; 1] For all x 2 A 1 , x  1 ; and 1 2 A 1 , so maxA 1 = 1. For all x 2 A 1 ; x > 0 , but 0 2 = A 1 , so minA 1 doesnít exist. The set of upper bounds of A 1 is the interval [1; + 1 [. The set of lower bounds of A 1 is the interval ]1; 0]. The smallest element among the upper bounds of A 1 is 1 , thus supA 1 = 1. The greatest element among the lower bounds of A 1 is 0 , thus inf A 1 = 0. 2- A 2 = fn 2 N : 5  n^2  25 g We Örst need to determine the elements of A 2. A 2 = f 3 ; 4 ; 5 g, and thus maxA 2 = 5; minA 2 = 3. The set of upper bounds of A 2 is the set f 5 ; 6 ; 7 ; :::g. The set of lower bounds of A 2 is the set f 0 ; 1 ; 2 ; 3 g. The least upper bound of A 2 is 5 (supA 2 = 5), and the greatest lower bound is 3 (inf A 2 = 3).

3- A 3 = fa 2 Z : a^2  36 g A 3 can be written as A 3 = fa 2 Z : a  6 g [ fa 2 Z : a  6 g: A 3 doesnít have a greatest element or a smallest element. This set is neither bounded above nor bounded below. Theorem 2 (The Completeness) Every non-empty subset of R that is bounded above has a least upper bound. Every non-empty subset of R that is bounded below has a greatest lower bound. The least upper bound and greatest lower bound of a subset of R, if they exist, are unique. Theorem 3 (Characterization of Supremum and InÖmum)

1. If A is an upper bounded set of R and M 2 R then

M = sup A ,

i) M is an upper bound of A ii) 8 " > 0 ; 9 x 2 A : x > M "

2. If A is a lower-bounded subset of R and m 2 R then

m = inf A ,

i) m is a lower bound of A ii) 8 " > 0 ; 9 x 2 A : x < m + "

Example 5. Consider the set A = un = 2 n n+1 ; n 2 N Each time we assign a value to n, we determine an element of A: For n = 1 we get u 1 = 3 2 A; A 6 = ?: Furthermore, for all n; we have: 2 < un  3 ((the sequence of the general term un = 2 n n+1 ; n 2 N (^) is decreasing and un! 2 if n! + 1 ): This demonstrates that A is bounded, and therefore, 9 sup A and 9 inf A: we have sup A = 3 2 A which implies max A = 3:On the other hand inf A = 2 2 = A; thus, minA does not exist. According to the characterization of the inÖmum, we know that :

inf A = 2 ,

i) 2 is a lower bound of A ii) 8 " > 0 ; 9 un 0 2 A : un 0 < 2 + "

4. Let a and b be two real numbers such that a  b.A semi-open and bounded interval in R is any set of the form : [a; b[ = fx 2 R : a  x < bg :

and also ]a; b] = fx 2 R : a < x  bg

5. Let a and b be two real numbers. A closed and unbounded interval in R is any set of the form :

[a; + 1 [ = fx 2 R : a  xg ; or

]1; b] = fx 2 R : x  bg :

6.We will denote the following special sets :

R+^ = [a; + 1 [ ; R += ]0; + 1 [ ; R^ = ]1; 0] ; R= ]1; 0[ :

The notation R^ denotes de set R excluding 0.

1.8.2 Neighborhood of a Point

The concept of a neighborhood will be used for the following chapters when we introduce the concept of limits. DeÖnition 7. (Neighborhood of a Point). Let a be a real number. We say that VR is a neighborhood of a if and only if there exists " > 0 such that : [a ",a + "]V.

1.8.3 Adherent Point - Accumulation Point - Isolated Point

Certain points play a speciÖc role in relation to certain subsets. DeÖnitions 8. Let E be a subset of R. We say that the point x 0 2 R is :

i. an adherent point of E if every open interval containing x 0 intersects E. ii. an accumulation point of E if every open interval containing x 0 intersects E at a point other than x 0. iii. an isolated point of E if it is adherent to E but is not an accumulation point of E. Remark 8. Every accumulation point of E is an adherent point of E: The set of adherent points is denoted by E and is called the closure ofE. The set of accumulation points is denoted by Eíand is called the derived set of E. Example 7. Let E =]0; 1[[f 2 g. The set of adherent points of Eis E = [0; 1] [ f 2 g: Every point x 2 ]0; 1[ is an accumulation point, and the set of accumulation points is E 0 = [0; 1]. The point x 0 = 2 2 E is an adherent point to E but not an accumulation point : it is an isolated point. Similarly, the points x 0 = 0 and x 0 = 1 are accumulation points and therefore adherent, but they do not belong to E.