= Solution
Let $R$ denote the number of rejected <null hypotheses> and $V$ the number of rejected true <null hypotheses>. The <familywise error rate> is $\Pr(V\ge1)$, whereas the <false discovery rate> is $\mathbb E[V/(R\vee1)]$, with a zero contribution when $R=0$. A valid true-null <p-value> satisfies $\Pr(p_i\le t)\le t$ for $0\le t\le1$.
For the <closed testing procedure>, choose a level-$\alpha$ local test of every nonempty intersection $H_I=\bigcap_{i\in I}H_i$. Reject $H_i$ only if every intersection containing it is rejected by its local test. If any true <null hypothesis> is rejected, the local test of $H_{I_0}$ must reject. Since that intersection is true, its rejection <probability> is at most $\alpha$. If $I_0$ is empty there can be no false rejection. This proves strong <closed-testing control of the familywise error rate>, under arbitrary dependence, provided each local test is valid under its whole intersection null.
For the <Benjamini-Hochberg procedure>, order the <p-values> and set
$$
R=\max\{r\in\{1,\ldots,m\}:p_{(r)}\le\alpha r/m\},
$$
with maximum zero if the set is empty; reject the first $R$ hypotheses. To prove its <false discovery rate> bound, fix a true-null index $i$. Replace $p_i$ by zero and let $R_i$ be the modified rejection count. This is a <function> of the other <p-values> and satisfies $R_i\ge1$.
The <Benjamini-Hochberg leave-one-out identity> is
$$
\frac{\mathbf1\{i\text{ rejected}\}}{R\vee1}=\frac{\mathbf1\{p_i\le\alpha R_i/m\}}{R_i}.
$$
Here is its deterministic justification. If $i$ was rejected, lowering its value cannot decrease $R$, and leaves all ordered values of ranks greater than $R$ unchanged. No such rank can newly satisfy its threshold, so $R_i=R$. Conversely, if $R_i=r$ and $p_i\le\alpha r/m$, at least $r-1$ other values and $p_i$ are at most this threshold. Therefore the original procedure has $R\ge r$. Monotonicity under decreasing $p_i$ also gives $R\le R_i=r$, so $R=r$ and $i$ is rejected. This establishes the identity; harmless deterministic tie-breaking also covers the artificial zero introduced in the modified data.
The assumed <independence> makes $p_i$ independent of $R_i$. Conditional on $R_i=r$, validity of the <p-value> implies
$$
\mathbb E\left[\left.\frac{\mathbf1\{p_i\le\alpha R_i/m\}}{R_i}\right|R_i=r\right]\le\frac1r\frac{\alpha r}{m}=\frac\alpha m.
$$
Sum over $i\in I_0$ to obtain
$$
\boxed{\operatorname{FDR}\le\frac{m_0}{m}\alpha\le\alpha.}
$$
If true-null <p-values> are exactly uniform rather than merely valid, the first inequality is equality. No restriction on dependence among the false-null <p-values> is needed beyond their joint independence from the true-null values.
For the requested <Simes inequality>, apply this result to the $m_0$ true-null <p-values> alone, for $m_0>0$. Every rejection is now false, so $V=R$ and the <false discovery rate> is the <probability> of any rejection. The event of at least one <Benjamini-Hochberg procedure> rejection is exactly
$$
\left\{\min_{1\le r\le m_0}\frac{p_{(r,I_0)}}r\le\frac\alpha{m_0}\right\}.
$$
Its <probability> is at most $\alpha$. This holds for $\alpha\in[0,1]$; with no true nulls the written minimum and division by $m_0$ are undefined and the error-control statement is instead vacuous.
The final step-up rule is the <Hochberg procedure>. Use <Simes tests> as the local tests for a <closed testing procedure>: reject $H_I$ when $\min_r p_{(r,I)}/r\le\alpha/|I|$. Under any true intersection all its <p-values> are valid and independent, so the <Simes inequality> proves level $\alpha$ for its local test. The implication supplied in the question says that whenever the <Hochberg procedure> rejects $H_{(i)}$, every intersection containing $(i)$ passes its local <Simes test>. Thus every such rejection is also a closed-testing rejection. Consequently
$$
\boxed{\operatorname{FWER}(\text{Hochberg})\le\operatorname{FWER}(\text{closed Simes tests})\le\alpha.}
$$
The true-null <independence> assumption is essential to this particular proof of local-test validity; the arbitrary-dependence guarantee of closed testing does not create valid local tests automatically.
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