Definition:Limit of 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:


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

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.