Proof.
Let c_1 = \max \{ c_ p, c_{p + 1}\} , where c_ p, c_{p +1} are the integers found in Lemma 69.22.3 for H^ p and H^{p + 1}. We will use this constant in the proofs of (1), (2) and (3).
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 our choice of c_1 we see that this is contained in I^{n - c_1}H^ p(X, \mathcal{F}) for n \geq c_1.
Note that part (3) implies part (2) by definition of the Mittag-Leffler condition.
Let us prove part (3). Fix an n throughout the rest of the proof. 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, I^ n\mathcal{F}) \ar[r] & 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}) \\ H^ p(X, I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^ p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^ a }
If m \geq c_1 we see that the image of a is contained in I^{m - c_1} H^{p + 1}(X, I^ n\mathcal{F}). By the Artin-Rees lemma (see Algebra, Lemma 10.51.3) there exists an integer c_3(n) such that
I^ N H^{p + 1}(X, I^ n\mathcal{F}) \cap \mathop{\mathrm{Im}}(\delta ) \subset \delta \left(I^{N - c_3(n)}H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right)
for all N \geq c_3(n). As H^ p(X, \mathcal{F}/I^ n\mathcal{F}) is annihilated by I^ n, we see that if m \geq c_3(n) + c_1 + n, then
\mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}))
In other words, part (3) holds with c_2(n) = c_3(n) + c_1 + n.
\square
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