Suppose we have $f: \mathbb{D} \rightarrow \mathbb{D}$ analytic and not identically zero. In order to prove $f$ has an infinite Blaschke product representation (where of course the product defines an analytic function on the disk), I need to show that $f$ necessarily has countably many zeros $a_j$ in $\mathbb{D}$, and that they satisfy $$\sum_{j=1}^{\infty} (1-|a_j|)<\infty.$$ This condition seems necessary because the Blaschke product formed by the zeros of $f$ converges locally uniformly on the disk if and only if they satisfy the summation condition, which is how one would show the product is analytic.
Assuming this crucial step, I am able to prove that $f$ has a representation $f(z)=B(z)g(z)$ where $B$ is a product as above and $g$ is a never vanishing analytic function. Any help is appreciated.