= Solution
For $r=1$, the assertion is the assumed vanishing for <coherent ideal sheaves>. For $r>1$, project $\mathcal F\subseteq\mathcal O_X^r$ onto the last component. Its image $\mathcal I\subseteq\mathcal O_X$ is a <coherent ideal sheaf>, and its <kernel> $\mathcal F'$ is a <coherent sheaf> contained in $\mathcal O_X^{r-1}$. Here images and kernels are coherent because a <variety> is <Noetherian>. The <short exact sequence>
$$
0\longrightarrow\mathcal F'\longrightarrow\mathcal F\longrightarrow\mathcal I\longrightarrow0
$$
gives an exact segment $H^1(X,\mathcal F')\to H^1(X,\mathcal F)\to H^1(X,\mathcal I)$ in the <long exact sequence in sheaf cohomology>. The outer terms vanish by <mathematical induction> and the hypothesis, so the middle term vanishes. This is <ideal-sheaf vanishing for a coherent submodule of a trivial bundle>.
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