Theorem. If the set $$\{a_0,\;a_1c,\; a_2c^2,\;\ldots\}$$ of the terms of a power series $$\sum_{n=0}^\infty a_nz^n$$ at the point $z = c$ is bounded, then the power series converges, even absolutely, for any value $z$ which satisfies $$|z| < |c|.$$

Proof. By the assumption, there exists a positive number $M$ such that $$|a_nc^n| < M \quad \forall\, n \,=\, 0,\,1,\,2,\,\ldots$$ Thus one gets for the coefficients of the series the estimation $$|a_n| < \frac{M}{|c|^n}.$$ If now $|z| < |c|$ , one has $$|a_nz^n| < M\left|\frac{z}{c}\right|^n,$$ and since the geometric series $\displaystyle\sum_{n=0}^\infty\left|\frac{z}{c}\right|^n$ is convergent, then also the real series $\displaystyle\sum_{n=0}^\infty|a_nz^n|$ converges.