= 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 of
$$
\binom n{k-1},\binom n{k-2},\ldots,\binom n1,1.
$$
Part (i) shows that replacing each $\binom nr$ by $\binom n{r+1}$ 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 gives
$$
f(n)=c_0\binom nk+c_1\binom n{k-1}+\cdots+c_{k-1}\binom n1+c_k
$$
with every $c_j\in\mathbb Z$.
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