Consider the inequality

Find the smallest integer for which this inequality holds, and prove that it holds for all larger integers.

**Claim:**The inequality

for all (and for no smaller positive integers ).

* Proof. * First, we show that it is not true for or . If , then we have

since implies , so the right side is larger than the left.

If , then we have

since again, , so . Therefore, the statement is false for both and .

Now, for we have

since . Hence, the statement is true for .

Now assume the statement is true for some .` So, we have,

Where the final inequality follows since and , so we have just made the term on the right smaller.

Thus, if the inequality is true for some , then it is true for . Hence, it is true for all

Why not use the converse of theorem I.19?