  Let $H^\infty(\mathbb D\times\N)$ be the Banach algebra of bounded
holomorphic functions defined on the disjoint union of countably many copies of
the open unit disk $\mathbb D\subset\mathbb C$. We show that the dense stable
rank of $H^\infty(\mathbb D\times\N)$ is one and using this fact prove some
nonlinear Runge-type approximation theorems for $H^\infty(\mathbb D\times\N)$
maps. Then we apply these results to obtain a priori uniform estimates of norms
of approximating maps in similar approximation problems for algebra
$H^\infty(\mathbb D)$.
