Definition 1. Let $X$ be a preordered set, $A \subseteq X$ and $a \in X$.
-
$a$ is an upper bound of $A$ if
\[\forall x \in A : x \leq a.\]$A$ is bounded from above if it has an upper bound.
The set of all upper bounds of $A$ is denoted by $\upper A$.
-
$a$ is a lower bound of $A$ if
\[\forall x \in A : a \leq x.\]$A$ is bounded from below if it has a lower bound.
The set of all lower bounds of $A$ is denoted by $\lower A$.
-
$a$ is bounded if it is both bounded from above and below.