In his book Munkres mentions that mathematicians have no qualms about using the finite axiom of choice:
Given a finite collection $\mathcal{A}$ of disjoint nonempty sets, there exist a set $C$ consisting of exactly one element from each element of $\mathcal{A}$.
However he dosen't give a proof of this fact (without using the axiom of choice). Is the following proof correct?
If $\mathcal{A}$ consists of only one set then we can pick any element $c$ from that set and let $C=\{c\}$.
Suppose the result holds for some $n$. We need to show that is holds also when $\mathcal{A}=\{A_1,\dots A_n, A_{n+1}\}$. By hypothesis there exsit a set $C$ consisting of exactly one element from each element of $\{A_1,\dots A_n\}$. Let $c$ be any element of $A_{n+1}$ and let $C'=C\cup\{c\}$. Then $C'$ contains exactly one element from each element of $\mathcal{A}$.
By induction we have shown that the result holds for any finite collection $\mathcal{A}$.