Number to Power of Zero Falling is One

Theorem

Let $x \in \R$ be a real number.

$x^{\underline 0} = 1$

where $x^{\underline 0}$ denotes the falling factorial.

Proof

 $\displaystyle x^{\underline 0}$ $=$ $\displaystyle \prod_{j \mathop = 0}^{-1} \left({x - j}\right)$ Definition of Falling Factorial $\displaystyle$ $=$ $\displaystyle 1$ Product is Vacuous

$\blacksquare$