Given a compact Metric Space
and a covering
by open sets, there exists a real number
such that every open ball of radius
is contained in some element of
.
The number
is called a Lebesgue number for the covering.
Proof:
Suppose that no Lebesgue number existed. Then there exists an open cover
such that for all
there exists an
such that
no 
contains 
. In
particular for each
we can choose a sequence
such
that
for
any
.
Since
is compact choose a convergent subsequence
with
for some
.
Since
is an open cover, we know there is some
and some
with
.
Again, as before, choose n such that
and
for
i 
n.
Check that
