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.

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. In fact the argument is by induction on the dimension of the support of $F$. We write a short exact sequence

$$0\to R(n)\to F(n-1)\to F(n)\to F\vert_H(n)\to 0$$ Note that $R$ is supported on a proper subvariety of $X$ since at the generic point $F(-1)=F$. Taking Euler characteristic, we get the result.

The above proof does not work in the ample case since an ample line bundle may not have any global sections (what is an example of this??).

How does one prove the result in the general case?