Open Ball is Neighborhood of all Points Inside

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

Let $B_\epsilon \left({a}\right)$ be an open $\epsilon$-ball of $a$ in $M$.

Let $x \in B_\epsilon \left({a}\right)$.

Then $B_\epsilon \left({a}\right)$ is a [Definition:Neighborhood (Metric Space)|neighborhoods]] of $x$ in $M$.

Proof
From Open Ball of Point Inside Open Ball:
 * $\exists \delta \in \R: B_\delta \left({x}\right) \subseteq B_\epsilon \left({a}\right)$

Thus by definition $B_\delta \left({x}\right)$ is a [Definition:Neighborhood (Metric Space)|neighborhoods]] of $x$ in $M$.