# Math Help - well orderable set, recursion

1. ## well orderable set, recursion

Suppose $C$ is a well orderable set, $f: C \times C \rightarrow C$, $A \subseteq C$, and let

$A_f =_{df} \cap \{ X \subseteq C | A \subseteq X \text{ }\& \text{ }f[X \times X] \subseteq X \}$

be the closure of $A$ under $f$. Define the sets $\{ A_n \}_{n \in \mathbb{N}}$ by the recursion

$A_0=A$, $A_{n+1}=A_n \cup f[A_n \times A_n]$,

and show that

$A_f=\cup_{n \in \mathbb{N}} A_n.$

I do not have any good ideas on how to prove this. It looks like this is true the way the sets are built up, but I do not see how to argue that. Thanks.