Equivalence of Definitions of Limit Point in Metric Space/Definition 3 implies Definition 1

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 $\alpha$ be a limit point in the topological space $\struct{S, \tau}$.

Then:

$\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$

Proof

From Open Ball is Open Set:

$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha$ is open set in $M$

By Definition of Topology Induced by Metric:

$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha$ is open set in $\struct{S,\tau}$

By Definition of Limit Point of Set:

$\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$

$\blacksquare$