Question

State the Axiom of Completeness.

Answer 1

Axiom of Completeness: "Every nonempty set of real numbers that is bounded above has a least upper bound."

The least-upper-bound property states that every nonempty set of real numbers having an upper bound must have a least upper bound(or supremum) in the set of real numbers.

The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers

{\displaystyle S=\{x\in \mathbf {Q} |x^{2}<2\}.}

This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be {\displaystyle {\sqrt {2}}}, but it does not exist in Q . For any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x.