Let $X_0$ be the set of open sets of $\mathbb{R}$. Define $X_{n+1}$ to be the set comprised of element of $X_n$ and their complement and countable unions. Then I cannot prove that $\bigcup_{n\in \omega} X_n$ is the Borel $\sigma$-algebra. I need to define $X_\alpha$ for $\alpha<\omega_1$ and then take the union to be able to prove that the result is a $\sigma$-algebra.
However, I don't know how to prove that $\bigcup_{n\in \omega} X_n$ is strictly smaller than the Borel $\sigma$-algebra either. An argument involving cardinality won't work here. Could you give me a hint or a reference?
Thanks in advance!
