Proof: Clearly for
it is obvious that
. And since
is increasing on that interval it stands to reason that
. Lastly noting that
. We may finally deduce that the proposed inequality is definitely true for
/ Therefore let us restrict our attention to
. On this interval
and since
is decreasing on this interval we see that
. We need the following lemma.
Lemma: Proof: Let
. Then
and
and since
the conclusion follows.
Using this lemma and the previous deductinos we see that
. Now evaluating
gives us
and the result follows.