[Vakil defines a scheme $X$ to be quasiseparated if the intersection of any two quasicompact opens is quasicompact]

This is part (b) of 7.3.C in Vakil's FOAG: Show that a morphism $\pi$ from a scheme $X$ into a scheme $Y$ is quasiseparated [ For any open affine $Spec(U)\subset Y$, $\pi^{-1}(Spec(U))$ is quasiseparated] if there is a cover of $Y$ by open affine subsets $U\_i$ such that $\pi^{-1}(U\_i)$ is quasiseparated.

Part (a) is to prove the same thing for quasicompactness and is straightforward. The hint suggests using the affine communication lemma and I can use that to show that for an arbitrary open subset $Spec(A)$ of $Y$, $\pi^{-1}(Spec(A))$ is finite union of quasiseparated open subsets, but that is clearly not sufficient.

Another approach seems to be to reduce the problem to showing that if $Spec(W)$ is an open subset of $\pi^{-1}(U\_i)$ and $Spec(V)$ is an open subset of $\pi^{-1}(U\_j)$, then $Spec(W)\cap Spec(V)$ is quasicompact. But i'm not even convinced that this is true in general.

Any hints or solutions would be greatly appreciated.