Closure via sequences

Proposition. Let $(X,d)$ be a metric space and let $A\subset X$. Then for $a\in X$,

Context. This is a specifically metric phenomenon (first-countability): the topological notion of closure can be detected by sequences.

Proof sketch.

• If $a\in\overline{A}$, then by ball intersections each ball $B(a;1/n)$ meets $A$; pick $a_n\in A\cap B(a;1/n)$ to get $a_n\to a$.
• Conversely, if $a_n\in A$ and $a_n\to a$, then every ball around $a$ contains some $a_n$, hence meets $A$, so $a\in\overline{A}$.