BND#DEF. Upper and Lower Bound.
Let $X$ be a preordered set, $A\subseteq X$ and $a\in X$.
$a$ is an upper bound of $A$ if for all $x\in A$,
\[x \leq a.\]
- $A$ 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 for all $x\in A$,
\[a \leq x.\]
- $A$ 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.
- Any greatest element is an upper bound.
- Any least element is a lower bound.