DEF-UPB. Upper and Lower Bound.
Let $(X,\preceq)$ be a preordered set, $A\subseteq X$ and $a\in X$.
$a$ is an upper bound of $A$ if
\[\forall x\in A : x\preceq a.\]$A$ is bounded from above if it has an upper bound.
$a$ is a lower bound of $A$ if
\[\forall x\in A : a\preceq x.\]$A$ is bounded from below if it has a lower bound.
$a$ is bounded if it is both bounded from above and below.
Any greatest element is an upper bound.
Any least element is a lower bound.