Proof.
Let $c = \max \{ c_ p, c_{p + 1}\} $, where $c_ p, c_{p + 1}$ are the integers found in Lemma 30.20.3 for $H^ p$ and $H^{p + 1}$.
Let us prove part (1). Consider the short exact sequence
\[ 0 \to I^ n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^ n\mathcal{F} \to 0 \]
From the long exact cohomology sequence we see that
\[ \mathop{\mathrm{Ker}}( H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}) ) = \mathop{\mathrm{Im}}( H^ p(X, I^ n\mathcal{F}) \to H^ p(X, \mathcal{F}) ) \]
Hence by Lemma 30.20.3 part (2) we see that this is contained in $I^{n - c}H^ p(X, \mathcal{F})$ for $n \geq c$.
Note that part (3) implies part (2) by definition of the Mittag-Leffler systems.
Let us prove part (3). Fix an $n$. Consider the commutative diagram
\[ \xymatrix{ 0 \ar[r] & I^ n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^ n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 } \]
This gives rise to the following commutative diagram
\[ \xymatrix{ H^ p(X, \mathcal{F}) \ar[r] & H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^ n\mathcal{F}) \ar[r] & H^{p + 1}(X, \mathcal{F}) \\ H^ p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u]^\gamma & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^\alpha \ar[r]^-\beta & H^{p + 1}(X, \mathcal{F}) \ar[u]_1 } \]
with exact rows. By Lemma 30.20.3 part (4) the kernel of $\beta $ is equal to the kernel of $\alpha $ for $m \geq c$. By a diagram chase this shows that the image of $\gamma $ is contained in the kernel of $\delta $ which shows that part (3) is true (set $k = n + m$ to get it).
$\square$
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