# Definition:Upper Bound of Sequence

*This page is about upper bounds of sequences which are bounded above. For other uses, see Definition:Upper Bound.*

## Definition

A special case of an upper bound of a mapping is an **upper 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 above in $T$ by $H \in T$.

Then $H$ is an **upper 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 above by $H \in \R$.

Then $H$ is an **upper bound of $\sequence {x_n}$**.