$\mathcal{S}$, smallest $\sigma$-algebra on $\mathbb{R}$ containing $\{(r,r+1):r\in\mathbb{Q}\}$, is the set of Borel subsets of $\mathbb{R}$
Show that $\mathcal{J}=\{[a, a+1) \mid a \in \mathbb{R}\}$ generates the Borel set of $\mathbb{R}$