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The Bonferroni correction tests each of $m$ hypotheses at level $\alpha /m$. The union bound then controls the familywise error rate by $\alpha$ under arbitrary dependence.

The Holm–Bonferroni method orders p-values and compares the $j$th smallest with $\alpha /(m-j+1)$, stopping at the first failed comparison. It controls the familywise error rate under arbitrary dependence.

Let $I_0=B^c$ and $m_0=|I_0|$. Each $q_j$ with $j\in I_0$ is a valid p-value by the argument in part c. If the Holm step-down procedure selects any index from $I_0$, let $r$ be the rank of the first such index. All $r-1$ earlier selections belong to $B$, soSelection through rank $r$ impliesConsequentlyThus the procedure controls the familywise error rate without requiring independence among the p-values.

Under the intersection null, every rejection is false, so the false-discovery proportion is $1_{\{R>0\}}$ and its expectation is the familywise error rate. Because the p-values are independent and exactly uniform, every super-uniform inequality in part (b) is an equality. Here $|I_0|=m$, and thereforeEquivalently, the order-statistic identity in the hint gives the same equality by induction on $m$.