For all prove that

From the first exercise of this section on inequalities, we know for all . But since for all and we have the inequality on the right immediately,

For the inequality on the left let

Then, we’ll consider the first two derivatives of to show that it is positive for all .

We know from the inequality on the right that for all . Hence, for all . Therefore, is increasing on the positive real axis. Since

we then have that for all . Hence, is increasing on the positive real axis, and since

we have that for all . Thus, for all ,