A Proof For 0!

I still remember:

YYK: "Mrs Kan, why is 0! = 1? Shouldn't it be zero?"
Mrs Pauline Kan: "Well, it's just by definition!"

So here's how you can actually show that 0! = 1; first of all, the '!' function is pronounced as factorial, and not just an exclaimation mark. So what does this function do? Well, easy, if you have a positive integer n, then n! simply means to do the following:

n! = n(n-1)(n-2)(n-3)...3(2)(1)

Well, so this means that n! is equal to the product of all positive integers preceding n. But you see, there's another way to define n!, and notice that:

n! = n(n-1)!

And then you must agree that:

1! = 1(0!)

But we all know that 1! = 1, and therefore we must agree that 0! = 1. :)