My book asks the reader to prove that the closure of $S$ (the intersection of all closed sets in $E$ that contain $S$) is the set of all limits of sequences of points in $S$ that converge in a metric space $E$. I believe I have a proof for this but am not sure of its formality:
① Take some sequence in $S$, $ ~~ a_1, a_2, a_3, \ldots ~ $ that converges to some $a \in E$.
② That is, for any $~\epsilon > 0$, $~~\exists N \in \mathbb{N} \mathrm{~~~s.t.~~} \forall n > N$, $~~ \mathrm{d}(a,a_n) < \epsilon$.
③ So by definition 4 of closed sets on Wolfram MathWorld, for $a$ to be outside of any closed set containing $S$ there would have to exist some ball of center $a$ that is disjoint from $S$.
④ Since $a$ is arbitrarily close to $S$ by ②, such a ball cannot exist, so $a \in \mathrm{closure}(S)$.
⑤ Conversely, any $b \in \mathrm{closure}(S)$ is the limit in $E$ of some convergent sequence in $S$.
Q.E.D.?
Thanks!