Let $\mathcal{S}$ be the smallest $\sigma$-algebra containing $\left\{(r,s]:r,s\in \mathbb{Q}\right\}$. Show that $\mathcal{S}$ is the collection of Borel sets.
It is already assumed that $\mathcal{S}$ is a $\sigma$-algebra. Do I just need to show that every open set in $\mathbb{R}$ is in $\mathcal{S}$? Do I need to show that half-open intervals are Borel sets? Or are there other things to consider as well? Any help is appreciated.