54-00-WhenAreBallsSeparated.tex

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\pmtitle{when are balls separated}
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 \usepackage{bbm}
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\newcommand*{\norm}[1]{\lVert #1 \rVert}
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\newtheorem{thm}{Theorem}
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\begin{document}
Let $(X,d)$ be a metric space, and let $B_r(x)$ be the $x$-centered
open 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
$$
    d(x,z)<r, \quad d(y,z)\le s,
$$
or
$$
    d(x,z)\le r, \quad d(y,z)< s.
$$
In either case, 
$$
   d(x,y)\le d(x,z)+d(z,y)<r+s.
$$
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\end{document}