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