If the metric space is compact and an open cover of is given, then there exists a number such that every subset of having diameter less than is contained in some member of the cover.
Such a number is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.
Proof
Let be an open cover of . Since is compact we can extract a finite subcover .
If any one of the 's equals then any will serve as a Lebesgue number.
Otherwise for each , let , note that is not empty, and define a function by .
Since is continuous on a compact set, it attains a minimum .
The key observation is that .
If is a subset of of diameter less than , then there exist such that , where denotes the ball of radius centered at (namely, one can choose as any point in ). Since there must exist at least one such that . But this means that and so, in particular, .