Closure of Subset of Metric Space by Convergent Sequence

Lemma
Let $M$ be a metric space.

Let $C \subseteq M$.

Let $x \in M$.

Then $x \in \operatorname{cl} \left({C}\right)$ iff there exists a sequence $\left \langle {x_n} \right \rangle$ in $C$ which converges to $x$.

Necessary Condition
Suppose there exists a sequence $\left \langle {x_n} \right \rangle$ in $C$ which converges to $x$.

Let $\epsilon > 0$.

Then by definition:
 * $\exists N \in \N: \forall n > N: x_n \in B_\epsilon \left({x}\right)$

where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$ in $M$.

Since $\forall n: x_n \in C$, it follows that:
 * $\forall \epsilon > 0: B_\epsilon \left({x}\right) \cap C \ne \varnothing$.

Hence $x \in \operatorname{cl} \left({C}\right)$.

Sufficient Condition
Now suppose $x \in \operatorname{cl} \left({C}\right)$.

By definition of closure:
 * $\forall n \in \N: \exists x_n \in C \cap B_{1 / n} \left({x}\right)$

Thus clearly $\left \langle {x_n} \right \rangle$ converges to $x$.