For and for
prove the following inequalities,
where we assume in the second inequality. Use this to show
Proof. From the previous exercise (Section 6.17, Exercise #41) we know
for . But by assumption we have
and
; hence,
. Therefore this inequality must hold for
:
Again, by the previous exercise we have
for . Since
and
this implies
. Therefore,
Letting , we have
I go through all the trouble of messing with binomial expansion… I feel like a chump.