Is $\mathcal B(\mathbb R) \times \mathcal B(\mathbb R)$ closed under finite intersection?

Question

Can anyone help me show that the Cartesian product of the Borel-sigma-algebra is closed under finite intersection?

Answer 1

Hint: Use that if $A,B,C,D$ are subsets of $\mathbb{R}$, then $(A\times B)\cap(C\times D)=(A\cap C)\times(B\cap D)$.