We will use the equivalence of categories of Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-i-star-equivalence} without further mention in the proof of the theorem. By Lemma \ref{lemma-fully-faithful} the functor is fully faithful. Thus we need to prove the functor is essentially surjective. \medskip\noindent Consider the collection $\Xi$ of quasi-coherent sheaves of ideals $\mathcal{K} \subset \mathcal{O}_X$ such that the statement holds for every object $(\mathcal{F}_n)$ of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ annihilated by $\mathcal{K}$. We want to show $(0)$ is in $\Xi$. If not, then since $X$ is Noetherian there exists a maximal quasi-coherent sheaf of ideals $\mathcal{K}$ not in $\Xi$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-acc-coherent}. After replacing $X$ by the closed subscheme of $X$ corresponding to $\mathcal{K}$ we may assume that every nonzero $\mathcal{K}$ is in $\Xi$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$. We will show that this object is in the essential image, thereby completing the proof of the theorem. \medskip\noindent Apply Chow's lemma (Lemma \ref{lemma-chow-noetherian-separated}) to find a proper surjective morphism $f : Y \to X$ which is an isomorphism over a dense open $U \subset X$ such that $Y$ is H-quasi-projective over $A$. Note that $Y$ is a scheme and $f$ representable. Choose an open immersion $j : Y \to Y'$ with $Y'$ projective over $A$, see Morphisms, Lemma \ref{morphisms-lemma-H-quasi-projective-open-H-projective}. Let $T_n$ be the scheme theoretic support of $\mathcal{F}_n$. Note that $|T_n| = |T_1|$, hence $T_n$ is proper over $A$ for all $n$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}). Then $f^*\mathcal{F}_n$ is supported on the closed subscheme $f^{-1}T_n$ which is proper over $A$ (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-proper} and properness of $f$). In particular, the composition $f^{-1}T_n \to Y \to Y'$ is closed (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). Let $T'_n \subset Y'$ be the corresponding closed subscheme; it is contained in the open subscheme $Y$ and equal to $f^{-1}T_n$ as a closed subscheme of $Y$. Let $\mathcal{F}_n'$ be the coherent $\mathcal{O}_{Y'}$-module corresponding to $f^*\mathcal{F}_n$ viewed as a coherent module on $Y'$ via the closed immersion $f^{-1}T_n = T'_n \subset Y'$. Then $(\mathcal{F}_n')$ is an object of $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. By the projective case of Grothendieck's existence theorem (Cohomology of Schemes, Lemma \ref{coherent-lemma-existence-projective}) there exists a coherent $\mathcal{O}_{Y'}$-module $\mathcal{F}'$ and an isomorphism $(\mathcal{F}')^\wedge \cong (\mathcal{F}'_n)$ in $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. Let $Z' \subset Y'$ be the scheme theoretic support of $\mathcal{F}'$. Since $\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1$ we see that $Z' \cap V(I\mathcal{O}_{Y'}) = T'_1$ set-theoretically. The structure morphism $p' : Y' \to \Spec(A)$ is proper, hence $p'(Z' \cap (Y' \setminus Y))$ is closed in $\Spec(A)$. If nonempty, then it would contain a point of $V(I)$ as $I$ is contained in the Jacobson radical of $A$ (Algebra, Lemma \ref{algebra-lemma-radical-completion}). But we've seen above that $Z' \cap (p')^{-1}V(I) = T'_1 \subset Y$ hence we conclude that $Z' \subset Y$. Thus $\mathcal{F}'|_Y$ is supported on a closed subscheme of $Y$ proper over $A$. \medskip\noindent Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent} the $\mathcal{O}_X$-module $\mathcal{H} = f_*\mathcal{F}'$ is coherent and by Lemma \ref{lemma-inverse-systems-push-pull} there exists a morphism $\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge$ in the category $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. Let $Z_0 \subset X$ be the scheme theoretic support of $\mathcal{H}$. It is clear that $|Z_0| \subset f(|Z'|)$. Hence $Z_0 \to \Spec(A)$ is proper (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}). Thus $\mathcal{H}$ is an object of  $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)$. Since each of the sheaves of ideals $\mathcal{K}^e$ is an element of $\Xi$ we see that the assumptions of Lemma \ref{lemma-existence-tricky} are satisfied and we conclude.