A scheme $X$ is locally noetherian iff there is a cover $X = \bigcup_i \text{Spec}(R_i)$ with noetherian $R_i$. When $X$ is also quasicompact, it is called noetherian.
Question: Why is (as Hartshorne states it) noetherian equivalent to $$ \textstyle(*)\hspace{1cm} \text{ There is a finite cover } X = \bigcup_{i=1}^n \text{Spec}(R_i) \text{ with noetherian } R_i ? $$
My attempt: It's clear that $X$ noetherian implies $(*)$. And it's also clear that $(*)$ implies $X$ locally noetherian, but why is $X$ quasicompact?
