Corollary 3

Tag 04VS, permanent. Cite it as [Bourbaki, Tag 04VS].

Commutative Algebra, chapter VI, § 7. Approximation theorem

Every valuation ring of $K$ containing $B$ contains one of the $A_i$ .

We may confine our attention to the case where $A_i \not\subset A_j$ for $i \# j$ . Let $V$ be a valuation ring of $K$ containing $B$ . We write

p = m(V) \cap B.

There exists a maximal ideal

p_i

B

containing

p

, whence

A_i = B_{p_i} \subset B, \subset V.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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