# Proper morphisms that are closed immersion on a fiber

I am interested in a morphism of $$S$$-schemes $$f : X \to Y$$ such that $$X$$ and $$Y$$ are proper over $$S$$ and there is some $$s_0 \in S$$ such that $$f : X_{s_0} \to Y_{s_0}$$ is a closed immersion. Is it true that there is an open neighborhood $$U \subset S$$ of $$s_0$$ so that $$X_U \to Y_U$$ is a closed immersion?

Because $$f$$ is automatically proper, I hoped to use the characterization of closed immersions as proper monomorphisms. Using an argument on formal neighborhoods of the fibers, I think I can show that $$f : X \to Y$$ is unramified at each point of the fiber and hence should be unramified on some neighborhood of the fiber. Maybe I can also show that $$f$$ is radicial in a similar fashion?

I am okay with assuming that $$X, Y$$ are noetherian and flat over $$S$$ if this helps.