Closed and unbounded set on a specific set or ordinals. I'm having problems on showing the following:
Let $\kappa$ be a strongly inaccesible cardinal. Show that the set of all ordinals $\alpha <\kappa$ such that $V_\alpha \models ZFC$ contains a club.
Any advice would be appreciated.
 A: We show that
$$
C := \{\alpha < \kappa \mid V_\alpha \prec V_\kappa \}
$$
is a club. As $V_\kappa$ is a model of ZFC, this proves the initial claim.
Note, that $C$ is closed (this follows from the Tarski-Vaught test).
To see that $C$ is unbounded, proceed as follows: Given $\alpha_0 < \kappa$, recursively construct a sequence $(\alpha_n \mid n \le \omega)$ as follows:
$\alpha_{n+1}$ is the least $\alpha < \kappa$ s.t. $V_{\alpha_n} \cup \{ \alpha_n \} \subseteq V_{\alpha}$ and s.t. for every formula of the form $\exists v_0 \phi(v_0,y_1, \ldots, v_n)$ and $x_1, \ldots, x_n \in V_{\alpha_n}$: If there is some $x_0 \in V_{\kappa}$ with 
$$V_{\kappa} \models \phi(x_0,x_1, \ldots, x_n),$$
then there already is some $x'_0 \in V_{\alpha}$ with
$$V_{\kappa} \models \phi(x'_0,x_1, \ldots, x_n).$$
Now $\alpha_\omega := \sup \{ \alpha_n \mid n < \omega \}$ satisfies
$$
V_{\alpha_\omega} \prec V_{\kappa}
$$
(again by the Tarski-Vaught test and the fact that $(\alpha_n \mid n < \omega)$ is strictly increasing).
