Let $\omega_A^\bullet$ be a normalized dualizing complex. Then $\delta^Y_Z = \delta$ is the dimension function associated with this dualizing complex. Observe that $\Ext^{-i}_A(M_n, \omega_A^\bullet)$ is a finite $A$-module annihilated by $I^n$. Fix $0 \leq i \leq s$. Below we will find $n_1 > n_0 > 0$ such that if we set $$ N = \Im(\Ext^{-i}_A(M_{n_0}, \omega_A^\bullet) \to \Ext^{-i}_A(M_{n_1}, \omega_A^\bullet)) $$ then the kernels of the maps $$ N \to \Ext^{-i}_A(M_n, \omega_A^\bullet),\quad n \geq n_1 $$ are finite length $A$-modules and the cokernels $Q_n$ form a system such that $\Im(Q_{n'} \to Q_{n''})$ has finite length for $n'' \gg n' \gg n_1$. This is equivalent to the statement that the system $\{\Ext^{-i}_A(M_n, \omega_A^\bullet)\}_{n \geq 1}$ is essentially constant in the quotient of the category of finite $A$-modules modulo the Serre subcategory of finite length $A$-modules. By the local duality theorem (Dualizing Complexes, Lemma \ref{dualizing-lemma-special-case-local-duality}) and Matlis duality (Dualizing Complexes, Proposition \ref{dualizing-proposition-matlis}) we conclude that there are maps $$ H^i_\mathfrak m(M_n) \to \Hom_A(N, E),\quad n \geq n_1 $$ as in the statement of the lemma. \medskip\noindent Pick $f \in \mathfrak m$. Let $B = A_f^\wedge$ be the $I$-adic completion of the localization $A_f$. Recall that $\omega_{A_f}^\bullet = \omega_A^\bullet \otimes_A A_f$ and $\omega_B^\bullet = \omega_A^\bullet \otimes_A B$ are dualizing complexes (Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing-localize} and \ref{dualizing-lemma-completion-henselization-dualizing}). Let $M$ be the finite $B$-module $\lim M_{n, f}$ (compare with discussion in Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-affine}). Then $$ \Ext^{-i}_A(M_n, \omega_A^\bullet)_f = \Ext^{-i}_{A_f}(M_{n, f}, \omega_{A_f}^\bullet) = \Ext^{-i}_B(M/I^n M, \omega_B^\bullet) $$ Since $\mathfrak m$ can be generated by finitely many $f \in \mathfrak m$ it suffices to show that for each $f$ the system $$ \{\Ext^{-i}_B(M/I^n M, \omega_B^\bullet)\}_{n \geq 1} $$ is essentially constant. Some details omitted. \medskip\noindent Let $\mathfrak q \subset IB$ be a prime ideal. Then $\mathfrak q$ corresponds to a point $y \in U \cap Y$. Observe that $\delta(\mathfrak q) = \dim(\overline{\{y\}})$ is also the value of the dimension function associated to $\omega_B^\bullet$ (we omit the details; use that $\omega_B^\bullet$ is gotten from $\omega_A^\bullet$ by tensoring up with $B$). Assumption (3) guarantees via Lemma \ref{lemma-elementary} that Lemma \ref{lemma-algebraize-local-cohomology-bis} applies to $B_\mathfrak q, IB_\mathfrak q, \mathfrak qB_\mathfrak q, M_\mathfrak q$ with $s$ replaced by $s - \delta(y)$. We obtain that $$ H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}(M_\mathfrak q) = \lim H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}( (M/I^nM)_\mathfrak q) $$ and this module is annihilated by a power of $I$. By Lemma \ref{lemma-terrific} we find that the inverse systems $H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}((M/I^nM)_\mathfrak q)$ are essentially constant with value $H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}(M_\mathfrak q)$. Since $(\omega_B^\bullet)_\mathfrak q[-\delta(\mathfrak q)]$ is a normalized dualizing complex on $B_\mathfrak q$ the local duality theorem shows that the system $$ \Ext^{-i}_B(M/I^n M, \omega_B^\bullet)_\mathfrak q $$ is essentially constant with value $\Ext^{-i}_B(M, \omega_B^\bullet)_\mathfrak q$. \medskip\noindent To finish the proof we globalize as in the proof of Lemma \ref{lemma-bootstrap}; the argument here is easier because we know the value of our system already. Namely, consider the maps $$ \alpha_n : \Ext^{-i}_B(M/I^n M, \omega_B^\bullet) \longrightarrow \Ext^{-i}_B(M, \omega_B^\bullet) $$ for varying $n$. By the above, for every $\mathfrak q$ we can find an $n$ such that $\alpha_n$ is surjective after localization at $\mathfrak q$. Since $B$ is Noetherian and $\Ext^{-i}_B(M, \omega_B^\bullet)$ a finite module, we can find an $n$ such that $\alpha_n$ is surjective. For any $n$ such that $\alpha_n$ is surjective, given a prime $\mathfrak q \in V(IB)$ we can find an $n' > n$ such that $\Ker(\alpha_n)$ maps to zero in $\Ext^{-i}(M/I^{n'}M, \omega_B^\bullet)$ at least after localizing at $\mathfrak q$. Since $\Ker(\alpha_n)$ is a finite $A$-module and since supports of sections are quasi-compact, we can find an $n'$ such that $\Ker(\alpha_n)$ maps to zero in $\Ext^{-i}(M/I^{n'}M, \omega_B^\bullet)$. In this way we see that $\Ext^{-i}(M/I^n M, \omega_B^\bullet)$ is essentially constant with value $\Ext^{-i}(M, \omega_B^\bullet)$. This finishes the proof.