Ideal-sheaf vanishing for a coherent submodule of a trivial bundle ($\mathcal{F}\subseteq {\mathcal{O}}_X^r⟹H^1(X,\mathcal{F})=0$)

If $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf, projection to the last coordinate makes any coherent $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ an extension of a coherent submodule of ${\mathcal{O}}_X^{r-1}$ by an ideal sheaf. The long exact sequence in sheaf cohomology proves the assertion by mathematical induction on $r$.