In this first paragraph of the proof we reduce the lemma to the case where $Y$ is of finite type over $\Spec(\mathbf{Z})$. We may and do replace the base scheme $S$ by $\Spec(\mathbf{Z})$. We can write $Y = \lim Y_i$ as a directed limit of separated algebraic spaces of finite type over $\Spec(\mathbf{Z})$, see Limits of Spaces, Proposition \ref{spaces-limits-proposition-approximate} and Lemma \ref{spaces-limits-lemma-descend-separated}. For all $i$ sufficiently large we can find a separated finite type morphism $X_i \to Y_i$ such that $X = Y \times_{Y_i} X_i$, see Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation} and \ref{spaces-limits-lemma-descend-separated-morphism}. Let $\eta_1, \ldots, \eta_n$ be the generic points of the irreducible components of $|X|$ ($X$ is Noetherian as a finite type separated algebraic space over the Noetherian algebraic space $Y$ and therefore $|X|$ is a Noetherian topological space). By Limits of Spaces, Lemma \ref{spaces-limits-lemma-topology-limit} we find that the images of $\eta_1, \ldots, \eta_n$ in $|X_i|$ are distinct for $i$ large enough. We may replace $X_i$ by the scheme theoretic image of the (quasi-compact, in fact affine) morphism $X \to X_i$. After this replacement we see that the images of $\eta_1, \ldots, \eta_n$ in $|X_i|$ are the generic points of the irreducible components of $|X_i|$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}. Having said this, suppose we can find a diagram $$ \xymatrix{ X_i \ar[rd] & X_i' \ar[l] \ar[d] \ar[r] & \mathbf{P}^n_{Y_i} \ar[ld] \\ & Y } $$ where $X_i' \to X_i$ is a $U_i$-admissible blowup for some dense open $U_i \subset X_i$ and the morphism $X_i' \to \mathbf{P}^n_{Y_i}$ is an immersion. Then the strict transform $X' \to X$ of $X$ relative to $X_i' \to X_i$ is a $U$-admissible blowing up where $U \subset X$ is the inverse image of $U_i$ in $X$. Because of our carefuly chosen index $i$ it follows that $\eta_1, \ldots, \eta_n \in |U|$ and $U \subset X$ is dense. Moreover, $X' \to \mathbf{P}^n_Y$ is an immersion as $X'$ is closed in $X_i' \times_{X_i} X = X_i' \times_{Y_i} Y$ which comes with an immersion into $\mathbf{P}^n_Y$. Thus we have reduced to the situation of the following paragraph. \medskip\noindent Assume that $Y$ is separated of finite type over $\Spec(\mathbf{Z})$. Then $X \to \Spec(\mathbf{Z})$ is separated of finite type as well. We apply Lemma \ref{lemma-chow-noetherian} to $X \to \Spec(\mathbf{Z})$ to find a dense open subspace $U \subset X$ and a commutative diagram $$ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & \Spec(\mathbf{Z}) & \mathbf{P}^n_\mathbf{Z} \ar[l] } $$ with all the properties listed in the lemma. Note that $Z$ has an ample invertible sheaf, namely $\mathcal{O}_{\mathbf{P}^n}(1)|_Z$. Hence $Z' \to Z$ is a H-projective morphism by Morphisms, Lemma \ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}. It follows that $Z' \to \Spec(\mathbf{Z})$ is H-projective by Morphisms, Lemma \ref{morphisms-lemma-H-projective-composition}. Thus there exists a closed immersion $Z' \to \mathbf{P}^m_{\Spec(\mathbf{Z})}$ for some $m \geq 0$. It follows that the diagonal morphism $$ X' \to Y \times \mathbf{P}^m_\mathbf{Z} = \mathbf{P}^m_Y $$ is an immersion (because the composition with the projection to $\mathbf{P}^m_\mathbf{Z}$ is an immersion) and we win.