Models of Comprehension Schema 
Let $M_\alpha$ for $\alpha\in ON$ be transitive sets and let $M=\bigcup_{\alpha\in ON}M_\alpha$. Suppose that (i) for every $\alpha<\beta$, we have $M_\alpha\in M_\beta$ and (ii) for every limit $\lambda$, $M_\lambda=\bigcup_{\alpha<\lambda} M_\alpha$. Prove that if for arbitrarily large $\alpha\in ON$, $M_\alpha$ satisfies the comprehension schema, then $M$ satisfies the Comprehension schema.

Any help would be appreciated
 A: The way I see it, you have to combine two observations, but actually
get away with weaker assumptions:
Proposition. Let $(M_{\alpha} \mid \alpha \in \operatorname{On})$ be a
sequence such that $M_{\alpha}$ is transitive,
$M_{\alpha} \in M_{\beta}$ for all
$\alpha < \beta \in \operatorname{On}$,
$M_{\lambda} = \bigcup \{ M_{\alpha} \mid \alpha < \lambda \}$ for
limit ordinals $\lambda$ and such that the set of $\alpha$'s such that
$M_{\alpha}$ satisfies comprehension for $\Sigma_{1}$ formulas is
unbounded. (We can get away with comprehension for $\Sigma_{0}$
formulas, if we know that our $M_{\alpha}$'s are closed under
pairing.) Then
$M := \bigcup \{ M_{\alpha} \mid \alpha \in \operatorname{On} \}$
satisfies full comprehension.
Proof. Let $\phi(v_{0}, \ldots, v_{n-1})$ be a formula in the language
of set theory with exactly $v_{0}, \ldots, v_{n-1}$ as free variables
and let $p_{1}, \ldots, p_{n-1} \in M$. We want to see that
$$\{x \in p_{1} \mid M \models \phi(x,p_{1}, \ldots, p_{n-1} \} \in M.$$
Fix $\alpha < \operatorname{On}$ such that
$\{p_{1}, \ldots, p_{n-1} \} \subseteq M_{\alpha}$ and such that
$\phi(v_{0}, p_{1}, \ldots, p_{n-1})$ is absolute between $M_{\alpha}$ and
$M$. (Such an $\alpha$ exists by the Reflection Principle, because
$(M_{\alpha} \mid \alpha \in \operatorname{On})$ is a strictly
increasing and continuous sequence of transitive sets.) Let
$\alpha < \beta \in \operatorname{On}$ be such that $M_{\beta}$
satisfies comprehension for $\Sigma_{0}$ formulas. Now, in
$M_{\beta}$, the statement
$$x \in p_{1} \wedge M_{\alpha} \models \phi(x,p_{1}, \ldots, p_{n})$$
can be expressed as a $\Sigma_{1}$ formula in paramaters
$p_{1}, \ldots, p_{n}, M_{\alpha}$. (Again, we actually can do a
little better. This formula can be set up in a way that it actually
does not depend on the formula $\phi$ itself, but only on its rank in
the Levy hierarchy.) Since it may also be expressed in a $\Pi_{1}$
way, we get (by the absoluteness of $\Delta_{1}$ formulas between
transitive structures, that
$$\{x \in p_{1} \mid M_{\alpha} \models \phi(x,p_{1}, \ldots, p_{n-1})
\} = \{x \in M_{\beta} \mid M_{\beta} \models \left( x \in p_{1}
  \wedge M_{\alpha} \models \phi(x, p_{1}, \ldots, p_{n-1}) \right)\}
\in M_{\beta} \subseteq M.$$ qed
