Direct 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's 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, since every
is contained in some
, the extreme value theorem shows
. Now we can verify that this
is the desired Lebesgue's number.
If
is a subset of
of diameter less than
, choose
as any point in
, then by definition of diameter,
, where
denotes the ball of radius
centered at
. Since
there must exist at least one
such that
. But this means that
and so, in particular,
.
Proof by contradiction
Suppose for contradiction that
is sequentially compact,
is an open cover of
, and the Lebesgue number
does not exist. That is: for all
, there exists
with
such that there does not exist
with
.
This enables us to perform the following construction:





Note that
for all
, since
. It is therefore possible by the axiom of choice to construct a sequence
in which
for each
. Since
is sequentially compact, there exists a subsequence
(with
) that converges to
.
Because
is an open cover, there exists some
such that
. As
is open, there exists
with
. Now we invoke the convergence of the subsequence
: there exists
such that
implies
.
Furthermore, there exists
such that
. Hence for all
, we have
implies
.
Finally, define
such that
and
. For all
, notice that:
, because
.
, because
entails
.
Hence
by the triangle inequality, which implies that
. This yields the desired contradiction.