In this work we prove in a constructive way a theorem of Rudin which says
that if $E$ is an analytic subset of the bidisc $D^2$ (with
multiplicities) which does not intersect a neighbourhood of the
distinguished boundary, then $E$ is the zero set (with multiplicities) of
a bounded holomorphic function. This approach allows us to generalize this
theorem and also some results obtained by P.S. Chee.