Math stuff and other things...but mostly math.: Quick proof

This is just something I randomly started thinking about, and came up with a short proof for:

Prove that for any multiplication of any positive integers greater than 1, that the product of these numbers and two will be greater than the product of the original numbers, with one of the numbers increased by 1 before multiplying.

Not too hard to show. Let our desired product be:
$a_1 \cdot a_2 \cdots a_n$

We wish to show that:

$a_1 \cdot a_2 \cdots a_n \cdot 2 \geq a_1 \cdot a_2 \cdots (a_n + 1)$

Divide both sides by most of the terms:

$a_n \cdot 2 \geq (a_n + 1)$

$a_n \geq 1$

Which was one of our givens. If you were proving this, you should probably show it by working forwards instead FROM the given, but I'm not doing that for you. It's easy to see that my steps are reversible.