Corollary 3
Tag 0CLO, permanent. Cite it as [Bourbaki, Tag 0CLO].
Commutative Algebra, chapter VIII, § 2. Dimension des algèbres
If A is of finite dimension, every nonzero finitely generated A-algebra is of finite dimension.
Indeed, one deduces from cor. 2, by induction on n, that the ring A[T₁, ..., Tₙ] is of finite dimension if A is of finite dimension; fortiori, every nonzero quotient of A[T₁, ..., Tₙ] is of finite dimension (§ 1, No. 3, prop. 6).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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