How can we show that $\sum\limits_{n=1}^\infty |c_n| < \infty$ when $c_n$ is taylor series coefficients of $f(z): \mathbb{C} \rightarrow \mathbb{C}$?

Question

I'm trying to understand a solution to old exam question. And i have trouble understanding and verifying (for myself) one of the steps.

if $f(z): \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic on $B(0,r)$ (ball centered at $0$ with radius $r < \infty$) and $\{c\}_n^\infty$ is the taylor series coefficients of $f(z)$ when $z\in B$, how can we show that $\sum\limits_{n=1}^\infty |c_n| < \infty$ (we also know from the original question that $f(0)=0$)

The author of the solution just states without elaborating that $\sum\limits_{n=1}^\infty |c_n| < \infty$ at one of the steps probably because its something trivial or a well known theorem .....

Can someone explain why the infinite sum of Taylor coefficients converge absolutely in conditions given above? or tell me the name of the theorem so I can check out the proof myself.

EDIT: after some hints from commentators I figured it out .