Definition:Bounded Above Sequence

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

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

Also see

 * Definition:Bounded Below Sequence
 * Definition:Bounded Sequence


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