What are instances of implicit reliance on countable or dependent choice in classic books? Two examples are

Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald

where it is claimed, in remark 1 after corollary 1.5, that a certain argument does not require the Axiom of Choice but in fact the argument does use Dependent Choice, as pointed out in this question (now deleted), and on page 42 in

P. Halmos, Measure Theory, Graduate Texts in Mathematics 18, Springer-Verlag, New York, 1974 (reprint of the edition published by Van Nostrand, New York, 1950)

where countable additivity of the Lebesgue measure is proved without mentioning Countable Choice (which is needed).

Note that I am not interested in opinions as to whether the authors were really aware of the implicit use and did not mention it, or were unaware of it; only the factual instances of such arguments not mentioning reliance on countable or dependent choice explicitly.