Lemma 28.29.3. Let X be a quasi-compact scheme. There exists a dense open V \subset X which is separated.
Proof. Say X = \bigcup _{i = 1, \ldots , n} U_ i is a union of n affine open subschemes. We will prove the lemma by induction on n. It is trivial for n = 1. Let V' \subset \bigcup _{i = 1, \ldots , n - 1} U_ i be a separated dense open subscheme, which exists by induction hypothesis. Consider
V = V' \amalg (U_ n \setminus \overline{V'}).
It is clear that V is separated and a dense open subscheme of X. \square
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