Let $f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$. Suppose that $f$ is holomorphic in $\mathbb{D}$ and $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ for $z \in \mathbb{D}$. Prove that if $f$ has a pole on the unit circle $\mathbb{T}$ then the above power series diverges at any $z \in \mathbb{T}$.
This question has been posted before and has a solution. My question is what if I don't assume that
$f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$.
The solution posted also does not use this hypothesis.
Can I assume that the statement is redundant? How does it affect the problem?