completeness property of real numbers I am studying about how a real number is defined by its properties. The three type of properties that make the real numbers what they are.


*

*Algebraic properties i.e, the axioms of addition, subtraction multiplication and division.

*Order properties i.e., the inequality properties

*Completeness property 
Here is the question : I am not able to understand the completeness property and please explain it to me in detail(it says upper bound lower bound ......)as I am a self learner.
 A: The crucial problem with rational numbers is that they are incomplete. It was discovered about 2300 years ago that there is no rational number whose square is 2.  But why should there be a square root of 2?  One reason is that it's easy to find a sequence of rational numbers that appears to be getting closer and closer to the square root of 2:
$$\frac11, \frac 32, \frac 75, \frac{17}{12}, \frac{41}{29},\ldots, \frac ab, \frac {a+2b}{a+b},\ldots$$ 
and one can show that although the terms in this sequence get closer and closer together, there isn't anything they get closer to, because if there was, its square would be $2$, and there is no rational number whose square is 2.  Or one can consider the sequence of rational numbers $$1, 1.4, 1.41, 1.414, 1.4142, \ldots$$ which is similar: the terms get closer and closer together as you look farther along the sequence, but they do not get close to any rational number, again because if they did that rational number would be a square root of 2, and there is no such rational.
So the rational numbers are literally incomplete; there are 
“missing” numbers.
The real numbers solve this problem: they are a system of numbers that contains the rationals, but has the property that if $S$ is any  sequence whose members eventually get closer and closer together, like the examples above, then there is some real number that the elements of $S$ approach as closely as desired; $S$ converges to some real number.  This is the “completeness” property you are looking for.
A: Suppose I have a set $X \subseteq \mathbb{R}$ such that for all $x \in X$ we have $x \le t$ for some number $t$. Then the completeness axiom guarantees the existence of a smallest number, call it $s$, such that $x \le s$.
You write this down more formally by saying that if $X$ is bounded, ie. if all $x$ are such that $x \le t$ for some $t$, then there exists a number $s \in \mathbb{R}$ satisfying the following:


*

*$x \le s$

*If $x \le t'$  then $s \le t'$
we call $s$ the supremum of $X$ and denote it by $\sup(X)$. 
The completeness axiom is declaring the existence of a supremum on a bounded set.
