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Let $X$ be a scheme covered by a finite number of affine open subsets $U_i$ such that for any $U_i, U_j$, the $U_i\cap U_j$ is a union of finite number of affine open subsets $W^{(i,j)}_h$. Then for any affine open subsets $U, V$, the $U\cap V$ is a union of a finite number of affine open subsets. This is essentially Vakil's note 6.1.H(p142).

It would be very appreciated if you give an elementary proof.(I knew the definition of schemes only a week ago. All I know is before 6.1.H.) Or any reference?

I want to show that projective schemes are quasi-separated. I know it is true if the above is true.

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    I think I c$a$n prove the last sentence about projective schemes without the above question. But I want the above question itself.2012-11-25

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Here is a hint: Skip forward in the notes to proposition 6.3.1. There it is proved that you can cover the intersection of two open affines by affine opens that are distinguished (that is $D(\cdot)$) in both of them.

Together with the observation that $U_i\cap U_j$ is quasicompact, I found that ingredient very useful in proving 6.1.H myself some time ago. Maybe try to reduce to the case that $X$ is covered by only two affines.

I would suggest you give the exercise another try. If you get too frustrated, you can still ask again for a solution or another hint.

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    Sorry in (4), $W_1,W_2$ should be replaced by $V_1,V_2$. In the post they are $W^{(i,j)}_h.$2012-11-26