At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathcal{B}(\Delta)$ is the Banach space of bounded, holomorphic functions on a domain $\Delta$.

**Lemma.** Let $\Delta$ be a domain in $\mathbb C^{m}$ and consider a bounded sequence $(f_k)_k$ of $\mathcal{B}(\Delta)$ satisfying the additional property that there exists some point $z_{0}\in\Delta$ such that the corresponding sequence of Taylor series $(\sum_n a_n ^k (z-z_0)^n)_k$ at $z_{0}$ is convergent in $\mathbb C [[z-z_0]]$ equipped with the projective topology (that is, each $(a_n ^k)_k$ converges in $\mathbb C$). Then $(f_{k})_{k}$ converges uniformly on compact sets of $\Delta$ towards some $f\in\mathcal{B}(\Delta)$.

This lemma is elementary to derive (I can include the proof if needed), so I doubt very much it is original. Yet I've never come across this statement in classical references or research articles I've been reading. Do you know a reference/name associated to it?

Thanks in advance!