# Definition:Bounded Above Sequence

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*This page is about sequences which are bounded above. For other uses, see Definition:Bounded Above.*

## Definition

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

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

Then $\left \langle {x_n} \right \rangle$ is **bounded above** if and only if:

- $\exists M \in T: \forall i \in \N: x_i \preceq M$

### 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.

Then $\sequence {x_n}$ is **bounded above** if and only if:

- $\exists M \in \R: \forall i \in \N: x_i \le M$

## Unbounded Above

$\left \langle {x_n} \right \rangle$ is **unbounded above** iff there exists no $M$ in $T$ such that:

- $\forall i \in \N: x_i \preceq M$

## Also see

- Definition:Bounded Above Mapping, of which a
**bounded above sequence**is the special case where the domain of the mapping is $\N$.