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Let $S$ be finite dimensional locally Noetherian regular scheme. Let $f \colon X \rightarrow S$ be locally of finite type. Then $f(X) \subset S $ contains a closed point of $S$? If it does, I'd like to know the proof.

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    $\begingroup$ Consider the case that $S$ is the spectrum of a DVR, and $X$ the spectrum of its field of fractions. $\endgroup$
    – Angelo
    Commented Jun 24, 2020 at 17:03

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