Let $X$ be a connected projective scheme. Let $U$ be a finite type open substack of the algebraic stack of coherent sheaves on $X$ with a fixed Hilbert polynomial. Can one take $p>0$ such that every sheaf in $U$ is Castelnuovo-Mumford $p$-regular? It seems true and used in some papers (maybe with additional condition), but I cannot show this statement. Could anyone give me a sketch of a proof?