Definition: For any $\displaystyle x \in \mathbb{R}$ and integer $\displaystyle k \geq 0$ we define $\displaystyle \binom{x}{0}=1$ and $\displaystyle \binom{x}{k}=\frac{x(x-1) \cdots (x-k+1)}{k!}, \ \ k \geq 1.$

Prove that for any integers $\displaystyle m,n,k,$ where $\displaystyle k \geq 0$ and $\displaystyle n \geq 1,$ we have: $\displaystyle \binom{\frac{m}{n}}{k} \in \mathbb{Z}[1/n].$