(1). We write
and
respectively with the objects which were formulated in
fact,
that is
-
and
-
Summing up yields
-
Here, the sum is again continuous in , with value .
(2). We start again with
-
and
-
and multiply both equations. This yields
Due to
fact
for
limits,
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
fact,
the left-hand factor converges for to , and because of the differentiability of in , the right-hand factor converges to .
(5) follows from (2) and (4).