  We show that the discrete Painlev\'e II equation with starting value
$a_{-1}=-1$ has a unique solution for which $-1 < a_n < 1$ for every $n \geq
0$. This solution corresponds to the Verblunsky coefficients of a family of
orthogonal polynomials on the unit circle. This result was already proved for
certain values of the parameter in the equation and recently a full proof was
given by Duits and Holcomb. In the present paper we give a different proof that
is based on an idea put forward by Tomas Lasic Latimer which uses orthogonal
polynomials. We also give an upper bound for this special solution.
