$\sigma$-algebra generate by uncountable collection of subsets. How would I approach this problem?

Let $\sigma(F)$ be a $\sigma$-algebra generated by $F$ where $F$ is an
uncountable collection of subsets of $\Omega$. If $A \in\sigma(F)$,
then there exists a countable subset $F_c$ of $F$ such that $A \in
 \sigma(F_c)$.


Use the "good set principle", i.e. Define
$$
M = \{A \in \sigma(F) \mid \exists \text{ countable } F_A \subset F \text{ such that } A \in \sigma(F_A)\}. 
$$
Show that this is a sigma algebra containing $F$. 
