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 $x_n$.
Some sources present $\ds \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.
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 theorem You 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.
- Results about limits of sequences can be found here.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.2$