We will use the equivalence of categories of Lemma \ref{lemma-i-star-equivalence} without further mention. For a closed subscheme $Z \subset X$ proper over $A$ in this proof we will say a coherent module on $X$ is ``supported on $Z$'' if it is annihilated by the ideal sheaf of $Z$ or equivalently if it is the pushforward of a coherent module on $Z$. By Proposition \ref{proposition-existence-proper} we know that the result is true for the functor between coherent modules and systems of coherent modules supported on $Z$. Hence it suffices to show that every object of $\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X)$ and every object of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ is supported on a closed subscheme $Z \subset X$ proper over $A$. This holds by definition for objects of $\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X)$. We will prove this statement for objects of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ using the method of proof of Proposition \ref{proposition-existence-proper}. We urge the reader to read that proof first. \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 Lemma \ref{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 supported on a closed subscheme $Z \subset X$ proper over $A$, thereby completing the proof of the theorem. \medskip\noindent Apply Chow's lemma (Lemma \ref{lemma-chow-Noetherian}) 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 $A$. 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}. Observe that $$ \text{Supp}(f^*\mathcal{F}_n) = f^{-1}\text{Supp}(\mathcal{F}_n) = f^{-1}\text{Supp}(\mathcal{F}_1) $$ The first equality by Morphisms, Lemma \ref{morphisms-lemma-support-finite-type}. By assumption and Lemma \ref{lemma-functoriality-closed-proper-over-base} part (3) we see that $f^{-1}\text{Supp}(\mathcal{F}_1)$ is proper over $A$. Hence the image of $f^{-1}\text{Supp}(\mathcal{F}_1)$ under $j$ is closed in $Y'$ by Lemma \ref{lemma-functoriality-closed-proper-over-base} part (1). Thus $\mathcal{F}'_n = j_*f^*\mathcal{F}_n$ is coherent on $Y'$ by Lemma \ref{lemma-pushforward-coherent-on-open}. It follows that $(\mathcal{F}_n')$ is an object of $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. By the projective case of Grothendieck's existence theorem (Lemma \ref{lemma-existence-projective}) there exists a coherent $\mathcal{O}_{Y'}$-module $\mathcal{F}'$ and an isomorphism $(\mathcal{F}')^\wedge \cong (\mathcal{F}'_n)$ in $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. Since $\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1$ we see that $$ \text{Supp}(\mathcal{F}') \cap V(I\mathcal{O}_{Y'}) = \text{Supp}(\mathcal{F}'_1) = j(f^{-1}\text{Supp}(\mathcal{F}_1)) $$ The structure morphism $p' : Y' \to \Spec(A)$ is proper, hence $p'(\text{Supp}(\mathcal{F}') \setminus j(Y))$ is closed in $\Spec(A)$. A nonempty closed subset of $\Spec(A)$ contains a point of $V(I)$ as $I$ is contained in the Jacobson radical of $A$ by Algebra, Lemma \ref{algebra-lemma-radical-completion}. The displayed equation shows that $\text{Supp}(\mathcal{F}') \cap (p')^{-1}V(I) \subset j(Y)$ hence we conclude that $\text{Supp}(\mathcal{F}') \subset j(Y)$. Thus $\mathcal{F}'|_Y = j^*\mathcal{F}'$ is supported on a closed subscheme $Z'$ of $Y$ proper over $A$ and $(\mathcal{F}'|_Y)^\wedge = (f^*\mathcal{F}_n)$. \medskip\noindent Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By Proposition \ref{proposition-proper-pushforward-coherent} the $\mathcal{O}_X$-module $\mathcal{H} = f_*(\mathcal{F}'|_Y)$ is coherent and by Lemma \ref{lemma-inverse-systems-push-pull} there exists a morphism $\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge$ of $\textit{Coh}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power $\mathcal{K}^t$ of $\mathcal{K}$. We obtain an exact sequence $$ 0 \to \Ker(\alpha) \to (\mathcal{F}_n) \to \mathcal{H}^\wedge \to \Coker(\alpha) \to 0 $$ in $\textit{Coh}(X, \mathcal{I})$. If $Z_0 \subset X$ is the scheme theoretic support of $\mathcal{H}$, then it is clear that $Z_0 \subset f(Z')$ set-theoretically. Hence $Z_0$ is proper over $A$ by Lemma \ref{lemma-closed-closed-proper-over-base} and Lemma \ref{lemma-functoriality-closed-proper-over-base} part (2). Hence $\mathcal{H}^\wedge$ is in the subcategory defined in Lemma \ref{lemma-systems-with-proper-support} part (2) and a fortiori in $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$. We conclude that $\Ker(\alpha)$ and $\Coker(\alpha)$ are in $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ by Lemma \ref{lemma-systems-with-proper-support} part (1). By induction hypothesis, more precisely because $\mathcal{K}^t$ is in $\Xi$, we see that $\Ker(\alpha)$ and $\Coker(\alpha)$ are in the subcategory defined in Lemma \ref{lemma-systems-with-proper-support} part (2). Since this is a Serre subcategory by the lemma, we conclude that the same is true for $(\mathcal{F}_n)$ which is what we wanted to show.