# Definition:Bounded Sequence/Metric Space

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*This page is about sequencess to metric spaces which are bounded. For other uses, see Definition: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.