# Definition:Bounded Below Set

## Definition

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

A subset $T \subseteq S$ is bounded below (in $S$) if and only if $T$ admits a lower bound (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 $\left({\R, \le}\right)$:

Let $\R$ be the set of real numbers.

A subset $T \subseteq \R$ is bounded below (in $\R$) if and only if $T$ admits a lower bound (in $\R$).

## Unbounded Below

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

A subset $T \subseteq S$ is unbounded below (in $S$) iff it is not bounded below.