Separated Morphism is Quasi-Separated

Theorem

Let $f$ be a separated morphism of schemes.

Then $f$ is quasi-separated.

Proof

Let $f$ be a separated morphism of schemes.

By definition, the diagonal morphism $\Delta_f$ is a closed immersion.

By Closed Immersion is Quasi-Compact $\Delta_f$ is quasi-compact.

Thus, by definition, $f$ is quasi-separated.

$\blacksquare$