[Proposition 3.3.1, BvdB]
Lemma 30.4.1 (Induction Principle). Let X be a quasi-compact and quasi-separated scheme. Let P be a property of the quasi-compact opens of X. Assume that
P holds for every affine open of X,
if U is quasi-compact open, V affine open, P holds for U, V, and U \cap V, then P holds for U \cup V.
Then P holds for every quasi-compact open of X and in particular for X.
Proof.
First we argue by induction that P holds for separated quasi-compact opens W \subset X. Namely, such an open can be written as W = U_1 \cup \ldots \cup U_ n and we can do induction on n using property (2) with U = U_1 \cup \ldots \cup U_{n - 1} and V = U_ n. This is allowed because U \cap V = (U_1 \cap U_ n) \cup \ldots \cup (U_{n - 1} \cap U_ n) is also a union of n - 1 affine open subschemes by Schemes, Lemma 26.21.7 applied to the affine opens U_ i and U_ n of W. Having said this, for any quasi-compact open W \subset X we can do induction on the number of affine opens needed to cover W using the same trick as before and using that the quasi-compact open U_ i \cap U_ n is separated as an open subscheme of the affine scheme U_ n.
\square
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