Definition:Bounded Sequence

Definition
A special case of a bounded mapping is a bounded sequence, where the domain of the mapping is $\N$.

Let $\struct {T, \preceq}$ be an ordered set.

Let $\sequence {x_n}$ be a sequence in $T$.

Then $\sequence {x_n}$ 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 $\struct {T, \preceq}$ is the set of real numbers under the usual ordering: $\struct {\R, \le}$: