  Let us consider two sequences of closed convex sets $\{A_n\}$ and $\{B_n\}$
converging with respect to the Attouch-Wets convergence to $A$ and $B$,
respectively. Given a starting point $a_0$, we consider the sequences of points
obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$
and $\{b_n\}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and
$a_n=P_{A_n}(b_n)$.
  Suppose that $A\cap B$ (or a suitable substitute if $A \cap B=\emptyset$) is
bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the
couple $(A,B)$ is $d$-stable, i.e., for each $\{a_n\}$ and $\{b_n\}$ as above
we have $\mathrm{dist}(a_n,A\cap B)\to 0$ and $\mathrm{dist}(b_n,A\cap B)\to
0$.
