Definition:Bounded Sequence/Metric Space

This page is about bounded sequence in the context of metric spaces. For other uses, see bounded.

Definition

Let $M$ be a metric space.

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

Then $\sequence {x_n}$ is a bounded sequence if and only if $\sequence {x_n}$ is bounded in $M$.

That is:

$\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$

Unbounded

$\sequence {x_n}$ is unbounded if and only if it is not bounded.