This is a part of the proof on a textbook of the fact that a power series that converges on an open disc defines an analytic function.
First note this inequality about a real series: for any $N,N^\prime\in\mathbb{N}$, $ \sum_{q=0}^{N^\prime}\sum_{p=0}^{N}\binom{p+q}{p}|a_{p+q}||b|^p|\zeta|^q\le\sum_{n=0}^\infty\sum_{p+q=n}\binom{p+q}{p}|a_{p+q}||b|^p|\zeta|^q=\sum_{n=0}^\infty|a_n|(|b|+|\zeta|)^n < \infty, $ where $|b|$ is less than the radius $r$ of the convergence of the series $\sum_{n=0}^\infty a_nz^n$, and $0 < |\zeta| < r - |b|$. Then, consider the complex series $\sum_{q=0}^{N}\sum_{p=0}^{N^\prime}\binom{p+q}{p}a_{p+q}b^p\zeta^q$, and decompose it into two like this: $ \left(\sum_{p+q\le N}+\sum_\text{other pairs}\right)\binom{p+q}{p}a_{p+q}b^p\zeta^q. $ Let $R$ be the second term. Here the textbook goes on to say that from the first inequality, for any given $\epsilon>0$, $|R|<\epsilon$ holds for sufficiently large $N$.
Why the last statement holds?