# Definition:Limit of Sequence/Metric Space

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## Contents

## Definition

Let $M = \left({A, d}\right)$ be a metric space or pseudometric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in A$.

Then $l$ is a **limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity**.

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

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

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.

## Also known as

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

## Also see

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.5$: Limits: Definition $5.2$