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 $i$.