Sequences - Numbers and Functions: Steps into Analysis - page 36

In relation to a sequence ( a ), the term 'eventually', the phrase 'provided

n is large enough', and the phrase 'for suMciently large n ' each mean
'for all n greater than some fixed number N '.
We can now gather the discussion into a formal definition and say that

+ ,

if and only if, given any number C , there is a number N such that
n N a n C.
Yet another way to express this definition is to say that, for each
number C , a

the sequence ( a n ) tends to

C for all but a finite number of terms of the sequence.

'( a ) tends to infinity' is often written ( a ) as n .

Archimedean order and the integer function

An important property of the real numbers is

given any number A , there is an integer n , which is greater than A
This is called the Archimedean property . It was first pointed out by the
Greeks, Archimedes among them, who probably wanted to say that if
you repeatedly combine together straight line segments of length 1, you
will never cover an area A , no matter how many times ( n times) you do
so. In contrast, when working with real numbers only, a number A can
always be exceeded by an integer.
By making our assumptions explicit and precise, we can see the
logical relations between different properties. For this reason you
should cite the Archimedean property, when you use it, at least in this
chapter. Later on, the Archimedean property can be assumed.
To see that the sequence of natural numbers tends to infinity we
argueas follows. If C is any number, by the Archimedean property, we
can choose a positive integer n such that C n . Then n n 1, so
C n 1. By induction all subsequent natural numbers are also
greater than C .So C is an eventual lower bound for the sequence of
natural numbers.
It also follows that there is an integer less than any given number
A : if we apply the Archimedean property to A there is an integer n
such that

n A . So any real number A divides the
set of integers into two sets, those less than or equal to A and those
greater than A . If for a given number A , m and n are integers such that
m A n , we need only scrutinise the n m integers, m ,
m 1, . . ., n 1 to determine the greatest integer less than or equal to
A . We denote this integer by [ A ].

A n , and so

[ ] 3, [2] 2 and [ 2.1] 3. [ A ] A [ A ] 1.