Definition:Bounded Ordered Set

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$ be both bounded below and bounded above in $S$.

Then $T$ is bounded in $S$.

Subset of Real Numbers
The concept is usually encountered where $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering: $\left({\R, \le}\right)$:

Also known as
A bounded ordered set can also be referred to as a bounded poset.