Borel sigma algebra not containing all subsets of $\mathbb{R}$?

Question

Consider the smallest sigma algebra $\mathscr{B}$ generated by all open subsets of $\mathbb{R}$. One would expect that $\mathscr{B}$ contains all subsets of $\mathbb{R}$, but as it turns out, if we assume the axiom of choice to be true, there are some subsets of $\mathbb{R}$ which don't belong to $\mathscr{B}$. I was hoping if someone could help me out with a proof of the above statement, i.e, $\mathscr{B}$ does not contain all subsets of $\mathbb{R}$.