On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as

a special case of Snapper's Lemma, see "An Intersection Theory for Divisors (preprint 1994)" by Steven Kleiman for a proof.

Kleiman's paper (or book?) mentioned by Nitsure must have either changed its title, never gone into publication, or elsehow disappeared, since I am unable to find it. Does anybody have a link, or an alternative source for the proof?

Thanks in advance.