DEF-SUP. Supremum and Infimum.
Let $(X,\leq)$ be a partially ordered set. Let $A\subseteq X$ and $a\in X$.
If the least upper bound of $A$ exists, it is called the supremum of $A$ and written as $\sup A$.
If the greatest lower bound of $A$ exists, it is called the infimum of $A$ and written as $\inf A$.