Is quasi-separated-ness needed for this induction principle?

Question

In stacks project Lemma 29.4.1 (08DR) an induction principle is stated for quasi-compact, quasi-separated schemes. But in the argument it seems that the quasi-separated-ness is not needed, and the quasi-compactness is sufficient.

Am I missing something? Or in the applications one simply does not need the induction principle for non-quasi-separated schemes?

Any help or reference or thoughts are appreciated. Thanks in advance.

Answer 1

Quasi-separatedness is needed since for the final induction argument to work, you need that the intersection $U_i\cap U_n$ is quasi-compact.