state the least upper bound axion for the real number state

Solution

An upper bound of a non-empty subset A of R is an element b R with b a for all a A.

M is an upper bound of A and if b is an upper bound of A then b M.

The real numbers satisfy the least upper bound property: any nonempty subset of the set of real numbers that is bounded from above has a least upper bound. This property does not hold if we replace the real numbers by the rational numbers.