let $S$ and $T$ be two disjoint compact nonempty sets. Show that there are points $x_0$ in $S$ and a point $y_0$ in $T$ such that $|x-y| \geq|x_0 -y_0|$ whenever $x$ is in $S$ and $y$ is in $T$.
Two Disjoint Compact sets
3
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metric-spaces
compactness
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0You should say in which space you're working. But since you use $|x-y|$ to denote the distance, I guess $S,T\subseteq\mathbb R$. – 2012-06-01
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0It might be useful to start by showing that the function $x\mapsto d(x,S)$ is continuous, see also [here](http://math.stackexchange.com/questions/8066/is-the-function-distance-continuous). The distance between a point $x$ and a set $S$ is defined by $d(x,S)=\inf_{s\in S} |x-s|$. – 2012-06-01