Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice)

Question

In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?

I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$.

Answer 1

Choice is not used to "select a half for which this is the case", because we can just pick the lower half if we need to make a choice.

Choice is not used to select an $a_{n_1}$ because we can pick the least $n$ such that $a_n \in I_1$. However, to select all of the $n_i$ together, we use the axiom of dependent choice (which is a very weak choice principle, but it implies countable choice).