when are balls separated

Let $(X,d)$ be a metric space, and let $B_r(x)$ be the $x$-centeredopen ball of radius $r$. If$d(x,y)\ge r+s$, then the balls$B_r(x)$ and $B_s(y)$ are separated.

To prove this, suppose that $B_r(x)$ and $B_s(y)$ are not separated.Then there exists a $z\in X$ such that either

or

In either case,