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.