calculus - Power Series and Absolute summability

I have a quick question regarding power series. Let $\psi_1, \psi_2, \ldots$ denote the real-valued coefficients of a power series. I would like to see a proof (or a counterexample) to the following result:

$$\left| \sum_{j=0}^{\infty} \psi_j z^j \right |< \infty \quad \forall \: z \in \mathbb{C} \textrm{ such that } |z| \leq 1 \implies \sum_{j=0}^{\infty} |\psi_j| < \infty.$$

If the result is true: could one replace the complex plane $\mathbb{C}$ by the real line $\mathbb{R}$?

Thanks very much for your help.