Prove that for , we have
and further that,
We prove each of these separately (and we use the first one in the proof of the second).
Proof #1. We prove each of the inequalities separately. First, we prove
This just requires some algebraic manipulation,
Since for all , we have the left inequality. For the right inequality we proceed similarly,
Thus, we have the inequality on the right true for all as well
Proof #2. Now, we want to use the result in the first proof to prove the second set of inequalities.
First, we consider the left inequality. For the case we have
on the left and right respectively. The inequality then holds since,
Thus, the inequality holds in the case . Assume then that the inequality holds for some . Then,
But, using the first part again we know ; hence,
Thus, the inequality is true for all .
Now, for the inequality on the right. For the case we have
on the left and right, respectively. But then, since we have,
Hence, the inequality is true for the case . Assume then that it is true for some . Then,
Hence the right inequality holds for all .
Therefore we have established both halves of the inequality for all