Definition:Bounded Sequence

Definition
A special case of a bounded mapping is a bounded 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 $T$.

Then $\left \langle {x_n} \right \rangle$ is bounded $\exists m, M \in T$ such that $\forall i \in \N$:
 * $(1): \quad m \preceq x_i$
 * $(2): \quad x_i \preceq M$

That is, it is bounded above and bounded below.

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