Talk:Lebesgue number lemma
From Maths
No, I do not think you can just choose the least out of a finite number of deltas.
Even if the covering is finite, still, some more effort is needed; and compactness must be used again.
One option: assume the opposite; take an infinite sequence of points that are worse and worse (that is, their relevant neighborhoods are smaller and smaller); by compactness, there exist an accumulation point of this sequence; now find a contradiction...
Another option: the maximal radius of a "good" neighborhood is a function of a point, and this function is continuous (and moreover, Lipschitz(1), think why); by compactness, it has the least value. Boris (talk) 17:39, 10 May 2016 (UTC)
