Supremum,Infimum,Archemidian property and the axiom of completeness

In my previous post i tried to discuss the bounds of sets of real numbers and also tried
to give some examples to describe what they really are.
but today i will be talking about supremum,infimum and other properties of bounded set.
Now we need to return to the topic of maxima and minima again.
in case you missed my last discussion on maxima and minima click here to be redirected back.
let $E$ be a set of real numbers having a maximum, say $M$, then we can describe that
maximum by the statement below:
$M$ is the least of all the upper bounds of $E$
this can also be described as $M$ is the least upper bound. now we must know that it is
possible for a set to have no maximum and yet be bounded.

Example
The open interval $(0,1)$ has no maximum, but has many upper bounds.
i know this is a little bit tricky but it is simple.
now we know that the interval $(0,1)=0<x<1$ so you can see that the number 0 and 1 are
not an element in the set.so certainly the interval has no maximum but many upper bounds
in the sense that $2$ is an upper bound and so is $1$. Now since the set has many upper
bounds greater than $1$ then it certainly means that $1$ is the least of all the upper
bounds.and we must note that $1$ cannot be described as a maximum because it fails to
be in the set.

lets give some definitions of the supremum and infimum.

supremum/least upper bound
let $E$ be a set of real numbers that is bounded above and non-empty, now if $M$ is the
least of all the upper bounds, then $M$ is said to be the least upper bound of $E$ or
supremum of $E$ written as $M=supE$.

infimum/greatest lower bound
Let $E$ be a set of real numbers that is bounded below and non-empty, if $m$ is the
greatest of all the lower bounds of $E$, then $m$ is said to be the greatest lower bound
of $E$ or the infimum of $E$ written as $m=infE$.

Now in order to complete the definition of $infE$ and $supE$ it is most convenient to be
able to write this expression even for $E=\phi$ or    for unbounded sets. thus written as

(1)$inf\phi=\infty$ and $sup\phi=-\infty$
(2)if $E$ is unbounded above, then $supE=\infty$
(3)if $E$ is unbounded below, then $infE=-\infty$

Axioms of Completeness
The axiom of completeness states that "A non-empty set of real numbers that is bounded
above has a least upper bound"
i.e If $E$ is non-empty and bounded above, then $supE$ exists and is a real number.

The Archimedean Property 
The archimedean property asserts that the set of natural numbers $\mathbb{N}$
has no upper bounds.
Now to understand this concept we must know that there is an important relationship
between the set of natural numbers $\mathbb{N}$ and the larger set of real numbers
$\mathbb{R}$.Because we have a well-formed mental image of what the set of reals
"looks like", this property is entirely intuitive and natural.It hardly seems that it
would require a proof.It says that the set of natural numbers has no upper bounds.

The Archimedean property is named after the famous Greek mathematician known as
Archimedes of Syracus(287 B.C-212 B.C).