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.