CH imples the existence of a function I was studying an article and the author stated that CH implies that there exists a function from $\omega_1 \setminus \omega$ onto the set of all countable subsets of $\omega_1$ such that for each $\omega\leq \alpha<\omega_1$, $F(\alpha)\subset \alpha$. He stated that it's a trivial consequence, but I can't see why. Can someone tell me how to prove the existence of this function?

Let $A=[\omega,\ \omega_1)=\{\alpha:\omega\le\alpha\lt\omega_1\}$ and let $B=\{X\subseteq\omega_1:X\text{ is countable }\}$. Using CH it is trivial to construct an injective function $g:B\to A$ with $X\subset g(X)$ for every $X\in B$; just list the elements of $B$ in a transfinite sequence of length $\omega_1$ and use transfinite recursion. Let $F$ be the inverse function of $g$, extended by setting $F(\alpha)=\emptyset$ whenever $\alpha$ is not in the range of $g$.
