Definition:Limit of Sequence/Metric Space

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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