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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 see you already found [this thread](http://math.stackexchange.com/questions/9809/quasiseparated-if-finitely-covered-by-affines-in-appropriate-way). Just thought to link it here.2012-11-25
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    Thank you. I didn't know how to link.2012-11-25
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    But I myself was not satisfied with that answer. It does not mean that it is bad. I cannot judge while it might be enough for advanced learners.2012-11-25
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    Dear Tom, If you want to show projective schemes are quasi-separated just as an exercise because you are learning the subject, why not invest more time in trying to solve the problem yourself instead of asking someone to solve it for you here; Vakil's notes are well-thought out and structured, and if he asks you to solve it at this point, he has given you enought tools to do so. On the other hand, if you need to know that projective schemes are quasi-separated as an ingredient in some other piece of work, than there are many references, e.g. in Hartshorne it is proved that they are ...2012-11-25
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    ... separated, which implies that they are quasi-separated. Regards,2012-11-25
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    Thank you for the reference. I know that Vakil's note is well organized. In fact I solved most exercises easily since page 1. That is why I am reading the note and why I believed that this problem may be easy also, then I am asking here. But your answer there was long and just a hint (containing undefined words), not an answer. If the proof is not so long, could you write it down please?2012-11-25
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    I think I can 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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