Stirling Number of the Second Kind of Number with Greater/Proof 1

Theorem
Let $n, k \in \Z_{\ge 0}$ such that $k > n$.

Proof
By definition, the Stirling numbers of the second kind are defined as the coefficients $\displaystyle \left\{ {n \atop k}\right\}$ which satisfy the equation:


 * $\displaystyle x^n = \sum_k \left\{ {n \atop k}\right\} x^{\underline k}$

where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.

Both of the expressions on the and  are polynomials in $x$ of degree $n$.

Hence the coefficient $\displaystyle \left\{ {n \atop k}\right\}$ of $x^{\underline k}$ where $k > n$ is $0$.