BOUNDS OF SETS OF REAL NUMBERS

In mathematics especially in the study of real numbers, bounds are widely used,but this bounds has a wide range of definition,for example we can give a simple definition of
bounds as "Assume $E$ to be some set of real numbers,there may or may not be a number $M$ that is bigger than every number in the set $E$.if there is, then we say that $M$ is the upper bound of the set $E$, if there exists no such upper
bounds then we say that the set is unbounded above or have no upper bounds.
same thing occurs when we try to talk of lower bounds,given any set E,
if there is an element say $m$ that is smaller than any other element in the set $E$,then we say the the set is bounded below in fact the
set is said to have a lower bound.
Now from the two definitions of upper and lower bounds we can get a definition for a bounded set.
A bounded set is simply defined as a set that have both upper and lower bounds.
as we can see this is a simple enough idea,but it is
critical to an understanding of the
real numbers and so we shall look more closely at it.

(1) Upper Bounds: Let $E$ be a set of real numbers.
if there exists a number say $M$
that is an upper bound for the set $E$ then
$x\leq{M}$ for all $x\in{E}$.
(2) Lower Bounds: let $E$ be a set of real numbers.if there exists a number $m$ that is a lower bound of the set $E$ then  $m\leq{x}$ for
all $x\in{E}$ Now that we now have a clear picture of what upper and lower bounds are. then we talk
about boundedness. we must note that boundedness is simply a set that has an upper and lower bounds.
A set can have many upper bounds. indeed we should understand that "every number is an upper bound for the empty set $\phi$" .
Now we proceed to giving a detailed explanation of what "maximum" and "minimum" means in relation to sets.
(1) Maximum: maximum is simply defined as the most naturally occurring upper bound among
the infinitely many to choose upper bounds, this is just the largest member of the set,and hence the
maximum.
in order words. Let $E$ be a set of real numbers.
if there exists a number $M$ that belongs to the set $E$ and is larger than any other member of $E$,then $M$ is called the maximum of the set $E$ and it is written as $M=maxE$.
(2) Minimum: Let $E$ be a set of real numbers. if there is a number $m$ that belongs to the set $E$ and it is the smallest of all the members of
$E$ then $m$ is called the minimum and denoted by $m=minE$.we will back up all our explanations
above with some examples that will shade more light on our topic of discussion.

Examples
(1)every closed interval [a,b],has a maximum and minimum and hence every closed interval is bounded.
for example if we define a closed set to contain [0,1] then we must understand that every closed set can be represented in the set builder notation
i.e $[0,1]=\{x:0\leq{x}\leq{1}\}$ since this is a closed interval then it certainly has a maximum and minimum and we should also understand that if a set has a maximum then that number is certainly the upper bound, same case for minimum. hence 0 is the minimum and lower bound of the set and 1 is the maximum and upper bound of the set.
hence the set is bounded since it has a lower and upper bound.
(2)every open interval has no maximum and no minimum and hence they are not bounded. Given an interval (0,1),this interval can also be denoted in the set builder notation i.e $(0,1)=\{x:0<x<1\}$ clearly we can see that the elements 0 and 1 are not members of the set and
hence we can conclude that the set has no maximum and minimum and so it does not have upper and lower bounds so the set is not bounded or it is an unbounded set.
(3)the set $\mathbb{N}$ of natural numbers has a minimum but no maximum, since it is a set that is
continuously counting. hence we can say that it is bounded below but not bounded above.
(4)the $\mathbb{Z}$ of integers has no lower or upper bounds, hence it is not a bounded set.