On page 4 of Nitin Nitsure's paper [Construction of Hilbert and Quot Schemes](https://arxiv.org/abs/math/0504590), the author refers to the fact that Hilbert polynomials are indeed polynomials as



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



Reference request: Kleiman's proof of Snapper's Lemma
A proof by Kleiman can be found in ‘‘[Toward a Numerical Theory of Ampleness](https://www.jstor.org/stable/pdf/1970447.pdf)’’. I suspect it's the intended proof, although the paper is from around 30 years before the cited 1994 preprint.


