# Definition:Lower Bound of Sequence

*This page is about lower bounds of sequences which are bounded below. For other uses, see Definition:Lower Bound.*

## Definition

A special case of a lower bound of a mapping is a **lower bound of a sequence**, where the domain of the mapping is $\N$.

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

Let $\left \langle {x_n} \right \rangle$ be a sequence in $T$.

Let $\left \langle {x_n} \right \rangle$ be bounded below in $T$ by $L \in T$.

Then $L$ is a **lower bound of $\left \langle {x_n} \right \rangle$**.

### Real Sequence

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

Let $\sequence {x_n}$ be a real sequence.

Let $\sequence {x_n}$ be bounded below in $T$ by $L \in \R$.

Then $L$ is a *lower bound of $\sequence {x_n}$'*.