In his book on abelian varieties, in the Appendix to section 6, Mumford says that if $X$ is any complete variety, $F$ is a coherent sheaf and $L$ an invertible sheaf, then the function defined as $$P(F,L,n)=\chi(F\otimes L^n)$$ is a polynomial in $n$. This is an exercise in Hartshorne, Chapter 3, 5.2, for a very ample line bundle.

I can see why the above is true for $L$ a very ample line bundle. However, it is not clear to me why this is true when $L$ is only assumed to be ample or is just given to be an invertible sheaf. Could someone please point out a proof.