Equivalence of Definitions of Limit Point in Metric Space

Theorem
Let $M = \struct {S, d}$ be a metric space.

Let $\tau$ be the topology induced by the metric $d$.

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.

$(1) \implies (2)$
Let:
 * $\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$

$(2) \implies (3)$
Let there exist a sequence $\sequence{\alpha_n}$ in $A$ such that $\alpha$ is a limit point of the sequence $\sequence{\alpha_n}$, considered as sequence in $S$.

$(3) \implies (1)$
Let:
 * $\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$