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Order $m$ p-values as $p_{(1)}\le \cdots \le p_{(m)}$ and take the largest $k$ with $p_{(k)}\le \alpha k/m$. The Benjamini-Hochberg procedure rejects the hypotheses corresponding to $p_{(1)},\dots ,p_{(k)}$ and controls the false discovery rate under independence and several forms of positive dependence.

Let $R$ be the number of rejections and $I_0$ the true-null indices. For $i\in I_0$, remove $p_i$ and let $R_i$ be the number determined by the corresponding leave-one-out step-up rule. On $\{i\text{ rejected},R=r\}$ one has $p_i\le \alpha r/m$ and $R_i=r$, while $R_i$ is independent of $p_i$. Hence super-uniformity givesSumming over $i\in I_0$ proves that the false discovery rate is at most $|I_0|\alpha /m\le \alpha$.