Given a separated scheme morphism $X\to Y$, and a morphism $Z \to Y$, Hartshorne proves that the extension $X\times_YZ \to Z$ is also separated, as long as the schemes involved are Noetherian. The valuative criterion he uses to prove it depends on this assumption.
Is it true for general $X, Y, Z$?
$X\to Y$ separated means that the diagonal map $X\to X\times_YX$ is a closed immersion. I have to use this somehow to prove that $$ \Delta:X\times_YZ \to (X\times_YZ)\times_Z(X\times_YZ) $$ is a closed immersion. Can that product be simplified in any way? Is there an obvious way to see this that I'm missing?