LUB property

Definition 1(Definition).

An ordered set $S$ is said to have the least upper bound property if for every subset $E$ of $S$, where $E$ is non empty and bounded above, $\sup E$ exists in $S$.

An analogous definition exists for the GLB property. Having the LUB property is the same thing as having the GLB property (LUB property $⟹$ GLB property: Consider the set of all lower bounds for a set $E$ that is non empty and bounded below; this is non empty and bounded above, and hence must have a supremum, which you can prove is the infimum of $E$. Similar reasoning works in the opposite direction).