I think it is probably easier to prove that $L$ is ample when all the inequalities are strict:

The assumption for $i=n$ implies that $K(L)$ is finite, i.e., $L$ is non-degenerate, by the second statement in the Riemann-Roch theorem in Mumford's "Abelian Varieties" and then the index theorem together with Riemman-Roch again (here you use the assumption for the other $i$ as well) imply that $L$ is ample.