Let \(\delta\) be a Lebesgue number of an open cover of a compact metric space. Which open ball, centered at any point \(x\), is guaranteed to lie inside some member?
The ball \(B(x,\delta/3)\) has diameter at most \(2\delta/3<\delta\), so it fits in a member. Larger balls can have diameter \(\delta\) or more and are not guaranteed.
A Lebesgue number of an open cover \(\mathcal{U}\) of a metric space is a \(\delta>0\) such that every subset \(A\) of diameter less than \(\delta\) fits inside some member of \(\mathcal{U}\).