Let $X$ be a Hausdorff space and suppose that $x\_n \rightarrow p$. Let $A$ be the set $\{x\_1,x\_2,\dots\}$, how to prove that $\overline{A}=A \cup \{p\}$.

The inclusion $A \cup \{p\} \subset \overline{A}$ is trivial. The other inclusion is easily proven in the case where $X$ is a metric space, [(as seen in this link)](https://math.stackexchange.com/questions/3377801/closure-of-convergent-sequence), but I have no idea how to prove the other inclusion in the general case.