Hartshorne book "Algebraic geometry" Proposition I.7.3 (b),
Proposition I.7.3.(b) : If $f: \mathbb Z \rightarrow \mathbb Z$ is any function, and $Q$ is a numerical polynomial such that $\Delta f=f(n+1)-f(n)=Q(n)$ for all $n\gg 0$, then there exists a numerical polynomial $P$ such that $f(n)=P(n)$ for all $n\gg 0$.
In proof of this proposition, If $\Delta(f-P)=0 \text{ for } n\gg 0$, then $(f-P)(n)$ is constant for $n\gg 0$ and so $f(n)=P(n)$ for $n\gg 0$.
But I can't understand this proof. May be I see more in detail.?