First true null argument for Holm control ($P_{(j)}\le \alpha /(m-j+1)\le \alpha /m_0$)

If the Holm step-down procedure makes a false rejection, it must reject the first true null in the ordered list. All $m_0$ true nulls remain at that step, so its cutoff is at most $\alpha /m_0$. The union bound on their marginally valid p-values gives familywise error rate at most $\alpha$. This proves strong control under arbitrary dependence, and super-uniformity suffices.