TABLE OF CONTENTS Let A be a nonempty set. We define
as the unique element M such that
If
then
In words, M is the least upper bound of the elements of A.
In an analogous manner, we define
as the unique element m such that
If
then
Put into words, m is the greatest lower bound of the elements A.
Let us consider a sequence of real numbers { x n}. Then
if for every ? > 0, there exists an integer N = N ( ?) such that
whenever n ? N.
The reader must avoid considering that, if the index n represents the time, as n gets larger and larger, the value of the sequence { x n} gets closer to x* [32]. Indeed, let us consider the following sequence:
This sequence tends to zero, but
Let { x n} be a series, and
If
it follows that
and
In an analogous manner, if m n = inf k ? n x k, it follows that m n ? m n+1, n ?0 and
Moreover, one can see that
and lim n ? ? x n exists if and only if
We get
which is usually written as follows:
In words, when the smallest cluster point (lim n ? ?
