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

For the second inequality, it’s important to not that to ).

Something got messed up in my reply. I meant to say that you must have or the last inequaliity is not true. The final step is valid only when . If , then the inequality would be reversed.

I go through all the trouble of messing with binomial expansion… I feel like a chump.