Center is Element of Closed Ball

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

Let $a \in A$.

Let $\epsilon \in \R_{>0}$ be a positive real number.

Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $M$.

Then:
 * $a \in \map {{B_\epsilon}^-} a$

Proof
By metric axiom $(\text M 1)$:
 * $\map d {a, a} = 0$

By assumption:
 * $\epsilon > 0$

Hence:
 * $\map d {a, a} < \epsilon$

By definition of the closed $\epsilon$-ball of $a$ $\map {{B_\epsilon}^-} a$ in $M$:
 * $a \in \map { {B_\epsilon}^-} a$