I have the following problem: Let $A$ be a set. If for every $\{x_1,x_2,...,x_n\} \subseteq A$ there exists a function $f_{x_1,...,x_n}: A \rightarrow A-\{x_1,...,x_n\}$. Then $A$ has a subset that in bijection with $\Bbb N$.
Aproach: Let $x_1 \in A$, let $x_2=f_{x_1}(x_1)$, then $x_3=f_{x_1,x_2}(x_2)$, and by induction, we can construct $x_k=f_{x_1,...,x_{k-1}}(x_{k-1})$ We define $g:\Bbb N \rightarrow A$ that maps $n \mapsto x_n$. It's easy to see that this map is injective, and taking the subset $g(\Bbb N)$ as image we get that $g$ is bijective.
But I'm not sure if this is correct. Specially if it's okay to restrict the image this way. Thanks in advance!
Yes, correct. I think this is like a book answer.