Distinct Points in Metric Space have Disjoint Open Balls/Proof 1

Proof
Let $\map d {x, y} = 2 \epsilon$.

Let $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ denote the open $\epsilon$-balls of $x$ and $y$ in $M$.

From Open Balls whose Distance between Centers is Twice Radius are Disjoint, $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ are the disjoint open $\epsilon$-balls whose existence we are to demonstrate.