Computing lebesgue number for an open covering

[Useful nucleus]

Munkres proof for the Lebesgue number lemma which is
(If X is a compact metric space and A is an open covering then there exists δ>0 such that for each subset of X having diameter less than δ , then there exists an element of the covering A containing it)
gives a way to compute δ using a finite subcollection that covers the compact metric space. However, this is not necessarily the smallest Lebesgue number. I wonder if there is another proof that involves evaluating the smallest Lebesgue number, if the latter exists.

 

[lugita15]

Useful nucleus, do you mean the largest Lebesgue number? Any positive number smaller than a Lebesgue number for a covering is also a Lebesgue number for the covering.

 

[Useful nucleus]

lugita15, you are absolutely right, I meant largest Lebesgue number if exist. I did not have time to refelct on this question again, but will give it a try soon.