For the ``usual'' proof of this fact, please see the discussion in the remark following this proof. We will prove this using Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving, see Lemmas \ref{lemma-spaces-diagonal}, \ref{lemma-spaces-RS-star}, and \ref{lemma-spaces-limits}. Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. \medskip\noindent Let $(R_n)$ be an inverse system of rings such that $R_n \to R_m$ is surjective with square zero kernel for all $n \geq m$. Let $X_n \to \Spec(R_n)$ be a finitely presented, flat, proper morphism where $X_n$ is an algebraic space and let $X_{n + 1} \to X_n$ be a morphism over $\Spec(R_{n + 1})$ inducing an isomorphism $X_n = X_{n + 1} \times_{\Spec(R_{n + 1})} \Spec(R_n)$. We have to find a flat, proper, finitely presented morphism $X \to \Spec(\lim R_n)$ whose source is an algebraic space such that $X_n$ is the base change of $X$ for all $n$. \medskip\noindent Let $I_n = \Ker(R_n \to R_1)$. We may think of $(X_1 \subset X_n) \to (\Spec(R_1) \subset \Spec(R_n))$ as a morphism of first order thickenings. (Please read some of the material on thickenings of algebraic spaces in More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-thickenings} before continuing.) The structure sheaf of $X_n$ is an extension $$ 0 \to \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0 $$ over $0 \to I_n \to R_n \to R_1$, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-deform}. Let's consider the extension $$ 0 \to \lim \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \lim \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0 $$ over $0 \to \lim I_n \to \lim R_n \to R_1 \to 0$. The displayed sequence is exact as the $R^1\lim$ of the system of kernels is zero by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-Rlim-quasi-coherent}. Observe that the map $$ \mathcal{O}_{X_1} \otimes_{R_1} \lim I_n \longrightarrow \lim \mathcal{O}_{X_1} \otimes_{R_1} I_n $$ induces an isomorphism upon applying the functor $DQ_X$, see Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-pullback-and-limits}. Hence we obtain a unique extension $$ 0 \to \mathcal{O}_{X_1} \otimes_{R_1} \lim I_n \to \mathcal{O}' \to \mathcal{O}_{X_1} \to 0 $$ over $0 \to \lim I_n \to \lim R_n \to R_1 \to 0$ by the equivalence of categories of Deformation Theory, Lemma \ref{defos-lemma-thickening-over-thickening-space-quasi-coherent}. The sheaf $\mathcal{O}'$ determines a first order thickening of algebraic spaces $X_1 \subset X$ over $\Spec(R_1) \subset \Spec(\lim R_n)$ by More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-first-order-thickening}. Observe that $X \to \Spec(\lim R_n)$ is flat by the already used More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-deform}. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-deform-property} we see that $X \to \Spec(\lim R_n)$ is proper and of finite presentation. This finishes the proof.