Boundary values of a convergent sequence of J-contractive matrix-functions

In this note it is proved that if $W_n(z)$ are $J$-contractive matrix-functions which are meromorphic in the disk $|z|<1$ ($J-W_n^*(z)JW_n(z)\ge0$, $J^*=J$, $J^2=I$), $W_n(z)\to W(z)$, as $n\to\infty$,
$$ W^*(z)JW(z)\le W_n^*(z)JW_n(z) $$
and
$$ \det W(z)\not\equiv0, $$
then there exists a subsequence $W_{n_k}(z)$ whose boundary values
$$ W^*_{n_k}(\zeta)JW_{n_k}(\zeta)\to W^*(\zeta)JW(\zeta)\quad (\text{a. e. }|\zeta|=1). $$
It follows from this result that every convergent Blaschke–Potapov product has $J$-unitary boundary values.