Definition of convergent sequence using only closure operator I have interest in Kuratowski closure axioms for topology. I would like to know how to define convergent sequence using only closure operator such that it is the same to definition of convergent sequence of equivalent axiomatic framework of topological spaces using open sets.
In this post: Why is a topology made of open sets? there is an answer about Kuratowski closure axioms for topology and the fact that using them many equivalent definitions can be made.
There, user "Vectornaut" states that "(WARNING: I'm kinda rusty at this, so these definitions may not be correct.)" and that "The sequence $\{ x_n \}$ converges to the point $x$ if $x$ touches every subsequence of $\{ x_n \}$." Translating this into closure operator axiomatic system (denoting closure operator by cl) we have that the sequence $\{ x_n \}$ converges to the point $x$ if $x$ is in the closure of every subsequence of $\{ x_n \}$. Is this statement true? Can someone maybe hint me towards some reference about proving equivalence of this with usual definition?

If $s: \mathbb{N} \to X$ is a sequence in $X$, we can define that $s$ converges to $x \in X$ by 
$$x \in \bigcap \{\operatorname{cl}(s[A]) : A \subseteq \mathbb{N} \text{ infinite }\}$$
which can be shown by considerations as William Elliott gave as well.
