Ring of rational integers is an interpolation domain

Statement

The ring of rational integers $\mathbb{Z}$ is an interpolation domain. More specifically, for any natural number $n$, there is a bijection between ${\mathbb{Z}}^{n+1}$ and the members of the ring of integer-valued polynomials of degree at most $n$, given as follows: the bijection sends a polynomial $f$ of degree at most $n$ to the $(n+1)$-tuple $\{f(0),f(1),\dots ,f(n)\}$.

In other words, for any $(n+1)$-tuple $(a_0,a_1,\dots ,a_n)\in {\mathbb{Z}}^{n+1}$, there is a unique polynomial $f\in \mathbb{Q}[x]$ that takes integers to integers, such that $f(i)=a_i$ for each $i$.