Exercise 6.1.G of Ravi Vakil's notes asks to prove that all affine schemes are quasi-separated, where quasi-separated schemes are defined as schemes where the intersection of any two quasi-compact open subsets is quasi-compact, or equivalently the "intersection of any two affine open subsets is a finite union of affine open subsets."
Can someone give a hint or solution?
The quasi-compact open subsets of $\mathrm{Spec}(A)$ have the form $\bigcup_i D(f_i)$ with finitely many $f_i \in A$. If we intersect two such sets, we obtain $\bigcup_i D(f_i) \cap \bigcup_j D(g_j) = \bigcup_{i,j} D(f_i g_j)$, which is a finite union of affine schemes, and therefore quasi-compact.