Exercise 2, § 4

Let $f : Z \to Z$ be a mapping. Prove that the following three properties are equivalent: a) There exist $n_0, n_1$ in $Z$ and $P$ in $Q[T]$ such that $f(n) = 0$ for $n \leq n_1$ and $f(n) = P(n)$ for $n \geq n_0$. b) There exists a finite support family $(a_i)_{i \in Z}$ of elements of $Z$ such that $f(n) = \sum_{i \in Z} a_i \binom{n + i}{i}$ for all $n \in Z$. c) There exist $r \in Z$ and $Q \in Z[T, T^{-1}]$ such that $\sum_{n \in Z} f(n) \; T^n = (1 - T)^r Q(T)$. One then says that $f$ is polynomial at infinity. Generalize to the case of mappings from $Z$ into an arbitrary $Z$-module.