Allow me to give you an indication of a relation that holds in very general settings. Consider a *first countable* topological space $(X, \mathscr{T})$, in other words a space in which every point $x \in X$ admits a *countable* neighbourhood base.

Given an arbitrary family $x$ of objects (sets, ultimately) indexed by set $I$ and an arbitrary subset $J \subseteq I$, let us write $x_J\colon=\{x_i\}_{i \in J}$ for the set of components of indices $i \in J$ of the family $x$. For arbitrary natural number $m \in \mathbb{N}$, let us denote the unbounded *natural* interval of all natural numbers $n \geqslant m$ by $[m, \rightarrow)$.

Given an arbitrary sequence of points $x \in X^{\mathbb{N}}$, it is not difficult to show that the set of limits of subsequences of $x$ is equal to the set of points *adherent* to the sequence $x$, set which is by definition expressed as $\displaystyle\bigcap_{n \in \mathbb{N}}\overline{x_{[n, \rightarrow)}}$. As an intersection of closed subsets, this subset is itself closed.

Metric spaces are in particular first countable (for any point $x \in X$ the collection $\left\{\mathrm{B}_q(x)\right\}_{q \in \mathbb{Q_{+}^{\times}}}$ of open balls centered at $x$ and of rational radius is a neighbourhood base), so the above statement particularly applies.