I am trying to prove that the countable, co-countable sigma algebra on $\mathbb{R}$ cannot be countably generated.
In more precise terms.
Let $\Sigma$ be the $\sigma$-algebra generated by countable subsets of $\mathbb{R}$, that is $$ \Sigma = \sigma (\{A\subseteq \mathbb{R} \:|\: A \textrm{ is countable}\})$$
It is easy to see that $A\in \Sigma$ iff $A$ is countable or co-countable.
Question: Is there a countable family $\{A_n\}_{n\in\mathbb{N}}$ such, for all $n\in\mathbb{N}$, $A_n\in \Sigma$ and
$$ \Sigma = \sigma (\{A_n \:|\: n\in\mathbb{N}\})?$$
I think the answer is NO, and I am trying to prove it. Can someone please help me in proving this?
My attempt is to prove by contradiction. That is assuming that the countable generating set exists then to show that sigma algebra generated by this set would miss some singletons of $\mathbb{R}$. Since the given sigma algebra contains all singletons this leads to contradiction. I am following this approach because I know that the set of all singletons generate the given sigma algebra and they are uncountable.