real number system 4

 We will return to discuss the axiom for completeness [41]. In general, given an arbitrary ordered field $A$, and a nonempty subset $M\in A$, if every $M$ which is bounded above always has a least upper bound $\sup M$, then it is said  "the field $A$ is complete."

This is the reason why the term of "completeness" is used in [41]. As we have accepted the assertion [41] as an axiom, real numbers become complete. But rational numbers are incomplete.

The completeness axiom [41] will be also written as below.

[41'] If a nonempty set $M$ of real numbers is bounded above, then there is a unique real number $\sup M $ such that
(1) $m\leq \sup M $ for all $m\in M$
(2) if $\epsilon>0$ (no matter how small), there is a $m\in M$ such that $m>\sup M - \epsilon$.

As real numbers are dense and a ordered field, $\sup M$ is unique (why?). (1) is the definition of $\sup M$. If $\sup M \in M$, (2) is also no doubt. Moreover (2) is true even though $\sup M\notin M$. We already have used (2) in the proof of Archimedean property.  (2) indicates that we should examine the cut in between  $M$ and $\sup M$.