The trouble here is that this is almost too obvious. The conclusion seems to follow immediately for natural numbers. So to make our proof seem more rigorous, let's take the long way around.

Proof

Let .

Suppose and . Then we have that and . We now have two numbers that are non-negative, therefore, their product must be non-negative. Thus we have:

.

Since , . Thus we make the left side of the last inequality even smaller if we subtract it. Hence,

as desired.

QED