 Solution

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 $|I_0|\alpha /m\le \alpha$.