We may assume that attains a local maximum in . This means that there exists an
,
such that
holds for all
.
Let be a sequence with
,
tending to
("from below“).
Then
,
and so
,
and therefore the difference quotient
-
Due to
fact,
this relation carries over to the limit, which is the derivative. Hence,
.
For another sequence with
,
we get
-
Therefore, also
and thus
.