Open Ball Contains Open Ball Less Than Half Its Radius

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

Let $a, b \in A$.

Let $\epsilon, \delta \in \R_{>0}$ such that $\delta \le \dfrac \epsilon 2$.

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$.

Let $\map {B_\delta} b$ be the open $\delta$-ball on $b$.

Let $a \in \map {B_\delta} b$.

Then:
 * $\map {B_\delta} b \subseteq \map {B_\epsilon} a$

Proof
We have: