Definition:Limit of a Sequence (Metric Space)

Definition

Let $\left({X, d}\right)$ be a metric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left({X, d}\right)$.

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

Then $l$ is known as the limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity and is usually written:

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

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.

From Sequence in Metric Space has One Limit at Most, it follows that the limit, if it exists, is unique.