# Definition:Limit of Sequence/Metric Space

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

Jump to navigation
Jump to search
## 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

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: This is a theoremYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

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 $5.2$