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 fact about::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 $$i$$.