If a compact set is covered by a finite union of open balls of same radii, can we always get a lesser radius? – math.stackexchange.com

If a compact set is covered by a finite union of open balls of same radii, can we always get a lesser radius? – math.stackexchange.com

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This question seems obvious, but I'm not secure of my proof. If a compact set $V\subset \mathbb{R^n}$ is covered by a finite union of open balls of common radii $C(r):=\bigcup_{i=1}^m B(c_i,r)$, ...

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