Suppose $H^\infty=\{f\in H(U):\exists M<\infty,\,\forall z\in U,\,|f(z)|\le M\}$ is the set of bounded holomorphic functions in the unit disk. It is a Banach algebra with pointwise multiplication.
Let $\varphi$ is an inner function, $f_n\in H^\infty$, $n=1,2,\ldots$, and $\varphi\cdot f_n\rightrightarrows g$. I am asked to show that $g/\varphi\in H^\infty$.
I tried Beurling factorization but did not succeed. The result is not true if $\varphi$ is not required to be an inner function. For example, $\varphi_0(z)=(z-1)^2$, the ideal generated by $\varphi_0$ is not closed.
Can anyone give a proof (or a hint)?