Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic \chi(L^k) of tensor powers of L is a polynomial in k. If L is very ample, this is just the Hilbert polynomial, and this can be proven by an induction argument twisting the short exact sequence 0 \to O(-1) \to O \to K \to 0. More generally, if L (or L*) is an ideal sheaf, the same argument should work. Why does the result still hold for arbitrary L?

Edit: I'd be particularly interested in an elementary proof that does not involve proving an entire Riemann-Roch theorem--Mumford is using this result to prove Riemann-Roch for abelian varieties!