 Solution

We use induction on $k$. If $k=0$, then $\delta f=0$, so the integer-valued function $f$ is constant. For $k>0$, the integer-valued function $\delta f$ satisfies ${\delta }^k(\delta f)=0$. By induction it is an integer linear combination ofPart (i) shows that replacing each $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ by $\left(\genfrac{}{}{0ex}{}{n}{r+1}\right)$ gives an integer-valued discrete antiderivative. Subtracting the resulting integer linear combination from $f$ leaves a function with zero forward difference, hence an integer constant. This giveswith every $c_j\in \mathbb{Z}$.