We have the following recursion principle on $\mathbb{N}$
Let $A$ be a non-empty set and $a_0 \in A$. Let $h : \mathbb{N} \times A \to A$ be a function. Then there is a unique function $f : \mathbb{N} \to A$ s.t. $f(0) = a_0$ and $\forall n \in \mathbb{N} \ f(n^{+})=h(n,f(n))$.
I'm trying to prove that any infinite $A \subseteq \mathbb{N}$ is equinumerous to $\mathbb{N}$ by constructing a bijection $f:\mathbb{N} \to A$ via the recursion principle.
What I want to do * is define
$f(0) = a_0$ for some $a_0 \in A$ and $\forall n \in \mathbb{N} \ f(n+) = \min\{ a \in A: a \notin \operatorname{range}(f|_n) \}$.
But what is $h$ in this example... $h(n,x) = ???$. Does anyone see how to define $h$ as to match the above construction?
*and this seems to be standard. Same construction is found in D.C. Goldrei's "Classic Set Theory".