The Limit of a Convergent Sequence in a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, and let $(x_{m})_{m\in\mathbb{N}}$ be a convergent sequence in $(X,d)$.

¶ The limit of $(x_{m})$, written $\lim_{m\to\infty}x_{m}$, is the point of $X$ to which $(x_{m})$ converges; it exists by Convergent Sequence in a Metric Space §convergent and is unique by Uniqueness of Limits in a Metric Space §unique.

¶ $(x_{m})$ converges to $\lim_{m\to\infty}x_{m}$ in $(X,d)$.