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

Theorem

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

Let $x, y \in M: x \ne y$.

Then there exist disjoint open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ containing $x$ and $y$ respectively.

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.

$\blacksquare$