At the begining of Vakil's FOAG, section 8.3.1, the author says
A morphism $\pi: X \rightarrow Y$ is quasiseparated if for every affine open subset $U$ of $\mathrm{Y}, \pi^{-1}(\mathrm{U})$ is a quasiseparated scheme (§5.1.1). (Equivalently, the preimage of any quasicompact open subset is quasiseparated, although we won't worry about proving this. This is the definition that extends to other parts of geometry.)
I tried to prove the equivalence of the two definitions but failed. It's easy to prove that the second implies the first definition, since affine scheme is quasiseparated. I found this too, but it seems not help (and even gives a new defition): Two definitions of quasi-separated morphisms which may not be equivalent?
So how can we prove that if for every affine open subset $U$ of $\mathrm{Y}, \pi^{-1}(\mathrm{U})$ is a quasiseparated scheme, then the preimage of any quasicompact open subset is quasiseparated? Thank you very much.