To prove the final assertion of the theorem we apply Local Cohomology, Proposition \ref{local-cohomology-proposition-annihilator} with $T \subset V(I) \subset \Spec(A)$. Namely, suppose that $\mathfrak p \not \in V(I)$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$. Then either there exists a prime $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ with $\mathfrak r \in V(I) \setminus T$ and we get $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s $$ by (4) in Situation \ref{situation-bootstrap} or there does not exist an $\mathfrak r$ and we get $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$ by Lemma \ref{lemma-helper-bootstrap}. In all three cases we see that $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s$. Thus Local Cohomology, Proposition \ref{local-cohomology-proposition-annihilator} (2) holds and we find that a power of $I$ annihilates $H^i_T(M)$ for $i \leq s$. \medskip\noindent We already know the other two assertions of the theorem hold for $i < s$ by Lemma \ref{lemma-final-bootstrap} and for the module $I^{m_0}M$ for $i = s$ and $m_0$ large enough. To finish of the proof we will show that in fact these assertions for $i = s$ holds for $M$. \medskip\noindent Let $M' = H^0_I(M)$ and $M'' = M/M'$ so that we have a short exact sequence $$ 0 \to M' \to M \to M'' \to 0 $$ and $M''$ has $H^0_I(M') = 0$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-divide-by-torsion}. By Artin-Rees (Algebra, Lemma \ref{algebra-lemma-Artin-Rees}) we get short exact sequences $$ 0 \to M' \to M/I^n M \to M''/I^n M'' \to 0 $$ for $n$ large enough. Consider the long exact sequences $$ H^s_T(M') \to H^s_T(M/I^nM) \to H^s_T(M''/I^nM'') \to H^{s + 1}_T(M') $$ Now it is a simple matter to see that if we have Mittag-Leffler for the inverse system $\{H^s_T(M''/I^nM'')\}_{n \geq 0}$ then we have Mittag-Leffler for the inverse system $\{H^s_T(M/I^nM)\}_{n \geq 0}$. (Note that the ML condition for an inverse system of groups $G_n$ only depends on the values of the inverse system for sufficiently large $n$.) Moreover the sequence $$ H^s_T(M') \to \lim H^s_T(M/I^nM) \to \lim H^s_T(M''/I^nM'') \to H^{s + 1}_T(M') $$ is exact because we have ML in the required spots, see Homology, Lemma \ref{homology-lemma-apply-Mittag-Leffler}. Hence, if $H^s_T(M'') \to \lim H^s_T(M''/I^nM'')$ is an isomorphism, then $H^s_T(M) \to \lim H^s_T(M/I^nM)$ is an isomorphism too by the five lemma (Homology, Lemma \ref{homology-lemma-five-lemma}). This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume that $H^0_I(M) = 0$. Choose generators $f_1, \ldots, f_r$ of $I^{m_0}$ where $m_0$ is the integer found for $M$ in Lemma \ref{lemma-final-bootstrap}. Then we consider the exact sequence $$ 0 \to M \xrightarrow{f_1, \ldots, f_r} (I^{m_0}M)^{\oplus r} \to Q \to 0 $$ defining $Q$. Some observations: the first map is injective exactly because $H^0_I(M) = 0$. The cokernel $Q$ of this injection is a finite $A$-module such that for every $1 \leq j \leq r$ we have $Q_{f_j} \cong (M_{f_j})^{\oplus r - 1}$. In particular, for a prime $\mathfrak p \subset A$ with $\mathfrak p \not \in V(I)$ we have $Q_\mathfrak p \cong (M_\mathfrak p)^{\oplus r - 1}$. Similarly, given $\mathfrak q \in T$ and $\mathfrak p' \subset A' = (A_\mathfrak q)^\wedge$ not contained in $V(IA')$, we have $Q'_{\mathfrak p'} \cong (M'_{\mathfrak p'})^{\oplus r - 1}$ where $Q' = (Q_\mathfrak q)^\wedge$ and $M' = (M_\mathfrak q)^\wedge$. Thus the conditions in Situation \ref{situation-bootstrap} hold for $A, I, T, Q$. (Observe that $Q$ may have nonvanishing $H^0_I(Q)$ but this won't matter.) \medskip\noindent For any $n \geq 0$ we set $F^nM = M \cap I^n(I^{m_0}M)^{\oplus r}$ so that we get short exact sequences $$ 0 \to F^nM \to I^n(I^{m_0}M)^{\oplus r} \to I^nQ \to 0 $$ By Artin-Rees (Algebra, Lemma \ref{algebra-lemma-Artin-Rees}) there exists a $c \geq 0$ such that $I^n M \subset F^nM \subset I^{n - c}M$ for all $n \geq c$. Let $m_0$ be the integer and let $m'(m)$ be the function defined for $m \geq m_0$ found in Lemma \ref{lemma-bootstrap} applied to $M$. Note that the integer $m_0$ is the same as our integer $m_0$ chosen above (you don't need to check this: you can just take the maximum of the two integers if you like). Finally, by Lemma \ref{lemma-bootstrap} applied to $Q$ for every integer $m$ there exists an integer $m''(m) \geq m$ such that $H^s_T(I^kQ) \to H^s_T(I^mQ)$ is zero for all $k \geq m''(m)$. \medskip\noindent Fix $m \geq m_0$. Choose $k \geq m'(m''(m + c))$. Choose $\xi \in H^{s + 1}_T(I^kM)$ which maps to zero in $H^{s + 1}_T(M)$. We want to show that $\xi$ maps to zero in $H^{s + 1}_T(I^mM)$. Namely, this will show that $\{H^s_T(M/I^nM)\}_{n \geq 0}$ is Mittag-Leffler exactly as in the proof of Lemma \ref{lemma-final-bootstrap}. Picture to help vizualize the argument: $$ \xymatrix{ & H^{s + 1}_T(I^kM) \ar[r] \ar[d] & H^{s + 1}_T(I^k(I^{m_0}M)^{\oplus r}) \ar[d] & \\ H^s_T(I^{m''(m + c)}Q) \ar[r]_-\delta \ar[d] & H^{s + 1}_T(F^{m''(m + c)}M) \ar[r] \ar[d] & H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r}) \\ H^s_T(I^{m + c}Q) \ar[r] & H^{s + 1}_T(F^{m + c}M) \ar[d] & \\ & H^{s + 1}_T(I^mM) } $$ The image of $\xi$ in $H^{s + 1}_T(I^k(I^{m_0}M)^{\oplus r})$ maps to zero in $H^{s + 1}_T((I^{m_0}M)^{\oplus r})$ and hence maps to zero in $H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r})$ by choice of $m'(-)$. Thus the image $\xi' \in H^{s + 1}_T(F^{m''(m + c)}M)$ maps to zero in $H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r})$ and hence $\xi' = \delta(\eta)$ for some $\eta \in H^s_T(I^{m''(m + c)}Q)$. By our choice of $m''(-)$ we find that $\eta$ maps to zero in $H^s_T(I^{m + c}Q)$. This in turn means that $\xi'$ maps to zero in $H^{s + 1}_T(F^{m + c}M)$. Since $F^{m + c}M \subset I^mM$ we conclude. \medskip\noindent Finally, we prove the statement on limits. Consider the short exact sequences $$ 0 \to M/F^nM \to (I^{m_0}M)^{\oplus r}/I^n (I^{m_0}M)^{\oplus r} \to Q/I^nQ \to 0 $$ We have $\lim H^s_T(M/I^nM) = \lim H^s_T(M/F^nM)$ as these inverse systems are pro-isomorphic. We obtain a commutative diagram $$ \xymatrix{ H^{s - 1}_T(Q) \ar[r] \ar[d] & \lim H^{s - 1}_T(Q/I^nQ) \ar[d] \\ H^s_T(M) \ar[r] \ar[d] & \lim H^s_T(M/I^nM) \ar[d] \\ H^s_T((I^{m_0}M)^{\oplus r}) \ar[r] \ar[d] & \lim H^s_T((I^{m_0}M)^{\oplus r}/I^n(I^{m_0}M)^{\oplus r}) \ar[d] \\ H^s_T(Q) \ar[r] & \lim H^s_T(Q/I^nQ) } $$ The right column is exact because we have ML in the required spots, see Homology, Lemma \ref{homology-lemma-apply-Mittag-Leffler}. The lowest horizontal arrow is injective (!) by part (5) of Lemma \ref{lemma-final-bootstrap}. The horizontal arrow above it is bijective by part (4) of Lemma \ref{lemma-final-bootstrap}. The arrows in cohomological degrees $\leq s - 1$ are isomorphisms. Thus we conclude $H^s_T(M) \to \lim H^s_T(M/I^nM)$ is an isomorphism by the five lemma (Homology, Lemma \ref{homology-lemma-five-lemma}). This finishes the proof of the theorem.