# Let $Y \subseteq B \subseteq A$ and $F[B] \subseteq B$. Prove by induction that $h(n) \subseteq B$ for all $n \in \omega$.

Let $$F:A\to A$$ be a function and let $$Y \subseteq A$$. The recursion theorem implies the existence of the function $$h:\omega \to \mathcal{P}(A)$$ that satisfies:

(1) $$h(0)=Y$$

(2) $$h(n^+)=h(n) \cup F[h(n)]$$

(a) Let $$Y \subseteq B \subseteq A$$ and $$F[B] \subseteq B$$. Prove by induction that $$h(n) \subseteq B$$ for all $$n \in \omega$$.
(b) Let $$C'=\bigcup_{n\in \omega}h(n)$$. Prove that $$Y \subseteq C' \subseteq A$$ and $$F[C'] \subseteq C'$$.
Attempt: For (a), I let $$I=\{n\in \omega |h(n) \subseteq B\}$$. $$0\in I$$ trivially. Then let $$n\in I$$, and we need $$(n^+)\in I$$, i.e. given $$h(n) \subseteq B$$, show that $$h(n^+)\subseteq B$$. I see the identity given in (2), but I'm not entirely sure how to proceed because the whole $$F[B]\subseteq B$$ thing is still confusing me. Any help is appreciated!