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