Convergence of $\sum_{n=0}^{\infty} a_n \overline{z}^n$ based on $\sum_{n=0}^{\infty} a_n z^n$. Let $\sum_{n=0}^{\infty} a_n z^n$ be a convergent series, where $\{a_n\}_{n \in \mathbb{N}} \in \mathbb{R}$ and $z \in \mathbb{C}$. Is $\sum_{n=0}^{\infty} a_n \overline{z}^n$ a convergent series?
This question popped into my mind as I was working on some homework for my analysis class - my gut says that it should be so (moreso I feel it should hold even if $\{a_n\} \in \mathbb{C}$ but that seems much more difficult to prove), but I am struggling to find a proof for this. 
I feel like there's some intuitive or clever way to show that the boundedness of $\sum_{n=0}^{\infty} a_n z^n$ implies the boundedness of $\sum_{n=0}^{\infty} a_n \overline{z}^n$, which is why I restrict the $\{a_n\}$s to reals, as any nice relationship I could find between the bounds of these series is obliterated when their coefficients are complex. 
I am particularly interested in either a proof or counterexample for the case of complex $\{a_n\}$s, but this problem has haunted my sleep for long enough and any proof of either real or complex $\{a_n\}$s would be more than sufficient. Any help would be greatly appreciated! Thanks so much in advance. 

Let $z=\displaystyle \sum_{n=0}^{\infty} a_nz^n$. By definition we have that for every $\varepsilon >0$, there exists $N \in \mathbb{N}$ such that $$\left \Vert \sum_{n=0}^m a_nz^n-z \right \Vert <\varepsilon, \: \forall m \geq N.$$ Note that $$\left \Vert \sum_{n=0}^m a_n\overline{z^n}-\overline{z} \right \Vert=\left\Vert \sum_{n=0}^m \overline{a_nz^n}-\overline{z} \right\Vert=\left \Vert \overline{\sum_{n=0}^m a_nz^n}-\overline{z} \right\Vert=\left\Vert \overline{\sum_{n=0}^m a_nz^n-z} \right\Vert=\left\Vert \sum_{n=0}^m a_nz^n-z \right\Vert.$$ Hence, for $m \geq N$ we get that $$\left\Vert \sum_{n=0}^m a_n\overline{z_n}-\overline{z}\right\Vert<\varepsilon.$$ So $\sum a_n \overline{z}^n$ converges to $\overline{z}$. 
Note that we used the property $\Vert w \Vert=\Vert \overline{w} \Vert$, for every $w \in \mathbb{C}$, where $\Vert w \Vert:= \sqrt{\Re w+\Im w}$ is the usual norm in the space of complex numbers. 
