Top-twist vanishing by hyperplane induction ($H^n({\mathbf{P}}^n,\mathcal{O}(m))=0(m\ge -n,n>0)$)

The hyperplane exact sequence for twisting sheaves identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on ${\mathbf{P}}^1$, propagates the degree-zero top vanishing down to $m=-n$. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to ${\mathbf{P}}^0$.