# Definition:Limit of Sequence/Metric Space

< Definition:Limit of Sequence(Redirected from Definition:Limit of Sequence (Metric Space))

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## Definition

Let $M = \struct {A, d}$ be a metric space or pseudometric space.

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

Let $\sequence {x_n}$ converge to a value $l \in A$.

Then $l$ is a **limit of $\sequence {x_n}$ as $n$ tends to infinity**.

If $M$ is a metric space, this is usually written:

- $\ds l = \lim_{n \mathop \to \infty} x_n$

## Also known as

A **limit of $\sequence {x_n}$ as $n$ tends to infinity** can also be presented more tersely as a **limit of $\sequence {x_n}$** or even just **limit of $x_n$**.

## Also see

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It can be seen that by the definition of open set in a metric space, this definition is equivalent to that for a limit in a topological space.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.2$