By Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover} we can choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$. For $p \geq 0$ set $$ W_p = U \times_X \ldots \times_X U \setminus \text{all diagonals} $$ where the fibre product has $p$ factors. Since $U$ is separated, the morphism $U \to X$ is separated and all fibre products $U \times_X \ldots \times_X U$ are separated schemes. Since $U \to X$ is separated the diagonal $U \to U \times_X U$ is a closed immersion. Since $U \to X$ is \'etale the diagonal $U \to U \times_X U$ is an open immersion, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-unramified} and \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}. Similarly, all the diagonal morphisms are open and closed immersions and $W_p$ is an open and closed subscheme of $U \times_X \ldots \times_X U$. Moreover, the morphism $$ U \times_X \ldots \times_X U \longrightarrow U \times_{\Spec(\mathbf{Z})} \ldots \times_{\Spec(\mathbf{Z})} U $$ is locally quasi-finite and separated (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fibre-product-after-map}) and its target is an affine scheme. Hence every finite set of points of $U \times_X \ldots \times_X U$ is contained in an affine open, see More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-over-affine}. Therefore, the same is true for $W_p$. There is a free action of the symmetric group $S_p$ on $W_p$ over $X$ (because we threw out the fix point locus from $U \times_X \ldots \times_X U$). By the above and Properties of Spaces, Proposition \ref{spaces-properties-proposition-finite-flat-equivalence-global} the quotient $V_p = W_p/S_p$ is a scheme. Since the action of $S_p$ on $W_p$ was over $X$, there is a morphism $V_p \to X$. Since $W_p \to X$ is \'etale and since $W_p \to V_p$ is surjective \'etale, it follows that also $V_p \to X$ is \'etale, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-local}. Observe that $V_p$ is a separated scheme by Properties of Spaces, Lemma \ref{spaces-properties-lemma-quotient-separated}. \medskip\noindent We let $U_p \subset X$ be the open subspace which is the image of $V_p \to X$. By construction a morphism $\Spec(k) \to X$ with $k$ algebraically closed, factors through $U_p$ if and only if $U \times_X \Spec(k)$ has $\geq p$ points; as usual observe that $U \times_X \Spec(k)$ is scheme theoretically a disjoint union of (possibly infinitely many) copies of $\Spec(k)$, see Remark \ref{remark-recall}. It follows that the $U_p$ give a filtration of $X$ as stated in the lemma. Moreover, our morphism $\Spec(k) \to X$ factors through $T_p$ if and only if $U \times_X \Spec(k)$ has exactly $p$ points. In this case we see that $V_p \times_X \Spec(k)$ has exactly one point. Set $Z_p = f_p^{-1}(T_p) \subset V_p$. This is a closed subscheme of $V_p$. Then $Z_p \to T_p$ is an \'etale morphism between algebraic spaces which induces a bijection on $k$-valued points for any algebraically closed field $k$. To be sure this implies that $Z_p \to T_p$ is universally injective, whence an open immersion by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open} hence an isomorphism and (1) has been proved. \medskip\noindent Let $x : \Spec(k) \to X$ be a quasi-compact morphism where $k$ is a field. Then the composition $\Spec(\overline{k}) \to \Spec(k) \to X$ is quasi-compact as well (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-quasi-compact}). In this case the scheme $U \times_X \Spec(\overline{k})$ is quasi-compact. In view of the fact (seen above) that it is a disjoint union of copies of $\Spec(\overline{k})$ we find that it has finitely many points. If the number of points is $p$, then we see that indeed $x \in T_p$ and the proof is finished.