Do you need the axiom of choice if the non-empty sets in question are defined recursively?

Question

Suppose I pick (using a single choice) an element $q_1$ from a nonempty set $U$. I then pick $q_2$ from the set $U/\{q_1\}$, $q_3$ from the set $U/\{q_1, q_2\}$ and so on. At each stage, the set in question is guaranteed to be non-empty. Thus, I have constructed a sequence $(q_n)_{n\in\mathbb{N}}$ using a single choice each time. Thus, I can safely assume that no (countably infinite) axiom of choice was used in the construction of this sequence, right?

Answer 1

You have in fact used a weak form of the axiom of choice, specifically, the axiom of dependent choice. Without choice you can make any finite number of arbitrary choices, but you cannot in general make an infinite sequence of them.