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.