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 $d(x,y)\\geq r+s$ , 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

d(x,z)<r,\\quad d(y,z)\\leq s,

d(x,z)\\leq r,\\quad d(y,z)<s.

In either case,

d(x,y)\\leq d(x,z)+d(z,y)<r+s.

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