Closed immersion on each fiber

Question

If $X\to Y$ is a map of proper $S$ schemes that is a closed immersion when restricted to each fiber over all $s\in S$, is $X\to Y$ a closed immersion? (Can assume $X$, $Y$ and $S$ are finite type over a field if necessary.)

Answer 1

This follows from the theorems from EGA IV quoted in this answer.

Namely, by 18.12.6 it is enough to show that $f: X \rightarrow Y$ is a monomorphism.

By 17.2.6 it is enough to show that the fiber $f^{-1}(y)$ of $f$ is either empty or isomorphic to $\text{Spec}(k(y))$, for each $y \in Y$.

But if $y$ is over $s \in S$, the fibers $f^{-1}(y)$ and $f_s^{-1}(y)$ are canonically isomorphic, so we are done by the assumption and using 17.2.6 again.