Open_Ball_of_Point_Inside_Open_Ball

Let $M = \left({A, d}\right)$ be a metric space. Let $B_\epsilon \left({x}\right)$ be an open $\epsilon$-ball in $M = \left({A, d}\right)$. Let $y \in B_\epsilon \left({x}\right)$. Then $\exists \delta \in \R: B_\delta \left({y}\right) \subseteq B_\epsilon \left({x}\right)$ That is, for every point in an open $\epsilon$-ball in a metric space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.