Equivalence of Definitions of Limit Point in Metric Space/Definition 2 implies Definition 3

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 $\alpha \in S$.

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 $S$.

Then:
 * $\alpha$ is a limit point in the topological space $\struct{S, \tau}$.