This is indeed the Hewitt-Marczewski-Pondiczery theorem. My proof, following Engelking, is [here][1]. It's in fact not that hard, the fact for a product of copies of 2 point discrete spaces already implies the general theorem pretty quickly.


  [1]: http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist;task=show_msg;msg=0487.0001