Define the Borel sigma-algebra on R^n as the smallest sigma-algebra containing all n-rectangles (a1, b1) x...x (an, bn).

Is it true that the Borel sigma algebra contains all sets of the form A1 x...x An, where each Ai is some Borel set in 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 ?