 Intermediate cohomology vanishing for a projective complete intersection

For a length-$r$ projective complete intersection $X_r\subseteq {\mathbb{P}}_k^N$, the groups $H^q(X_r,{\mathcal{O}}_{X_r}(m))$ vanish for every integer $m$ and $0<q<N-r$. Starting with cohomology of twisting sheaves on projective space, induct through $0\to {\mathcal{O}}_{X_{j-1}}(m-d_j)\to {\mathcal{O}}_{X_{j-1}}(m)\to {\mathcal{O}}_{X_j}(m)\to 0$. Both adjacent groups in the long exact sequence in sheaf cohomology vanish precisely in the indicated range.