An open cover of \([0,1]\) has Lebesgue number \(\delta=0.12\). What is the smallest \(n\) such that, when \([0,1]\) is split into \(n\) equal subintervals, each is guaranteed to lie within some member?
Each subinterval has length \(1/n\) and must satisfy \(1/n<0.12\), i.e. \(n>8.33\). For \(n=8\) the length \(0.125\) is not less than \(\delta\); for \(n=9\), \(1/9\approx0.111<0.12\).
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 \(\mathcal{U}\).