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 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:

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