The Stacks project

Proof. We only sketch the proof and omit some of the details. Set $X_1 = X_{red}$. Define $X_ n \subset X$ as the subfunctor defined by the rule: a morphism $f : T \to X$ where $T$ is a scheme factors through $X_ n$ if and only if the $n$th power of the ideal sheaf of the closed immersion $X_1 \times _ X T \to T$ is zero. Then $X_ n \subset X$ is a subsheaf as vanishing of quasi-coherent modules can be checked fppf locally. We claim that $X_ n \to X$ is representable by schemes, a closed immersion, and that $X = \mathop{\mathrm{colim}}\nolimits X_ n$ (as fppf sheaves). To check this we may work étale locally on $X$. Hence we may assume $X = \text{Spf}(A)$ is a Noetherian affine formal algebraic space. Then $X_1 = \mathop{\mathrm{Spec}}(A/\mathfrak a)$ where $\mathfrak a \subset A$ is the ideal of topologically nilpotent elements of the Noetherian adic topological ring $A$. Then $X_ n = \mathop{\mathrm{Spec}}(A/\mathfrak a^ n)$ and we obtain what we want. $\square$