This is a question I have regarding this discussion of the GIT Rigidity Lemma.
Suppose $p: X \to S$ is a flat morphism of schemes, where $S$ consists of only one point (i.e. $S = \operatorname{Spec}(A)$ where $A$ is an artinian local ring). Also suppose $H^0(X_s, \mathcal O_{X_s}) = k(s)$, where $X_s$ is the fiber of $p$ over $s$ and $k(s) = A / \mathfrak m_s$ is the residue field of $S$ at $s$, and suppose that there is a section $\eta: S \to X$ of $p$.
In the linked discussion there is the claim that $p$ is a proper map. Why is that the case? Any help would be appreciated :)