Define the Borel sigma-algebra on $\mathbb{R}^n$ as the smallest sigma-algebra containing all $n$-rectangles $(a_1, b_1) \times \cdots \times (a_n, b_n)$.

Is it true that the Borel sigma algebra contains all sets of the form $A_1 \times \cdots \times A_n$, where each $A_i$ is some Borel set in $\mathbb{R}$?

I thought this would be trivially true, but I had a lot of trouble trying to prove it, and I'm not even sure its true anymore.

If this is a well-known result, could you please refer me to a text where it has been (dis)proved ?