## radius of convergence equivalency in complex analysis

Originally Posted by problem statement
Let $\sum c_nz^n$ have a radius of convergence $R:=\sup\{|z|:\sum c_nz^n\text{ converges}\}$. Prove that $R=s$, where

$s:=\sup\{|z|:c_nz^n\to 0\text{ as }n\to\infty\}$.
Of course it's easy to show that $R\leq s$, but I'm having problems showing that $s\leq R$. Any help would be much appreciated.

Thanks!