Open Ball contains Strictly Smaller Closed Ball

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

Let $a \in A$.

Let $\epsilon, \delta \in \R_{\gt 0}$ such that $\epsilon < \delta$.

Let $\map {B^-_\epsilon} a$ be the closed ball centered on $a$ of radius $\epsilon$.

Let $\map {B_\delta} a$ be the open ball centered on $a$ of radius $\delta$.

Then:
 * $\map {B^-_\epsilon} a \subseteq \map {B_\delta} a$

Proof
By definition of subset:
 * $\map {B^-_\epsilon} a \subseteq \map {B_\delta} a$