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
-
![{\displaystyle {}{\frac {f(s_{n})-f(a)}{s_{n}-a}}\geq 0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae1c38fa6da0fbe687c5c22a1c94bf5489b14bd)
Due to
fact,
this relation carries over to the limit, which is the derivative. Hence,
.
For another sequence
with
,
we get
-
![{\displaystyle {}{\frac {f(t_{n})-f(a)}{t_{n}-a}}\leq 0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4e6df68ceebbe6848e7f0df227e01bbab3e20f)
Therefore, also
and thus
.