TITLE: Countable axiom of choice: why you can't prove it from just ZF
QUESTION [15 upvotes]: This is a follow-up question to the discussion about the finite axiom of choice here.
Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in that discussion, we can prove the existence of a choice function $c_n$ such that $c_n(i)$ belongs to $A_i$ for $i = 1,\cdots,n$
Again reasoning as suggested and using induction, we can prove the existence of a function $C$ defined over $\{1, 2, \cdots\}$ such that $C(n)$ is a choice function defined for $i=1,\cdots,n$ with $C(n)(i)$ an element of A_i. We can require in addition that $C(n)(i)$ and $C(n-1)(i)$ are equal for $i=1,\cdots,n-1$.
Now it would it seem that we can prove the countable axiom of choice by taking the union of $C(1)$, $C(2)$, $\cdots$
The question is as follows: why is that proof wrong? I suspect the answer may have something to do with the axiom of union.

REPLY [2 votes]: My motivation for asking this question was to understand if the standard proof of the Bolzano-Weierstrass theorem used axiom of choice in any way. If S is an infinite subset of [0,1], BW asserts the existence of a limit point. It is proved by repeatedly halving the interval and picking the half which contains infinitely points in S, much like Konig's lemma in Carl Mummert's reply.
As a result of the informative replies above, it is completely clear that the proof of BW does not use the axiom of choice in any way. The key point is that each new interval in the sequence can be defined as a function of the previous interval with a representative predicate. Therefore ordinary induction suffices. 
On the other hand, one may use the axiom of dependent choice to prove that any infinite set contains a countably infinite subset. Even the weaker axiom of countable choice suffices (see comment below). In fact, the statement about every infinite set containing a countably infinite subset is even weaker than the countable axiom of choice (see comment below). 
Thanks to Asaf Karagila, Andrej Bauer, and most of all, Carl Mummert for clarifying the role of the axiom of dependent choice. Thanks to Andreas Blass for pointing out that the coherence requirement in the original argument serves no purpose. Thanks to Hurkyl for pointing out that the original argument makes so many choices as to exhaust all the available free will in a world without the axiom of choice!