(1). We write
respectively with the objects which were formulated in
Summing up yields
Here the sum is again continuous in with value .
(2). We start again with
and multiply both equations. This yields
the expression consisting of the last six summands is a continuous function with value for
(3) follows from (2), since a constant function is differentiable with derivative .
(4). We have
Since is continuous in due to
the left hand factor converges for to and because of the differentiablity of in the right hand factor converges to .
(5) follows froms (2) and (4).