Definition:Limit of Sequence/Metric Space

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

 * Convergent Sequence in Metric Space has Unique Limit