Prove that for ,

and

*Proof #1.*Assume . Then, if we have

Where implies so multiplying both sides by preserves the inequality. So, the statement is true for . Assume then that the statement is true for some . Then,

Thus, the statement holds for ; and hence, for all

* Proof #2. * Assume . Then, for the case we have

since still allows us to preserve the inequality after multiplying both sides by . So, the inequality holds for the case . Assume then that it holds for some . Then,

Thus, the inequality holds for ; and hence, for all