Let $\mathbb{C}[[z]]$ be the set of formal power series over $\mathbb{C}$ with the usual definition of multiplication and addition making it a commutative ring with 1. Denote by $z\mathbb{C}[[z]]$ the ideal consisting of the series with constant term equal to 0.
Suppose that $a_{m}(z)= \sum_{n\geq 0} a_{mn}\cdot z^{n}$ is a sequence of formal power series in $z$, such that $a_{m}(z)\in z^{m}\mathbb{C}[[z]]$ for all $m$, that is the first non-zero coefficient of each $a_{m}(z)$ is the coefficient of $z^{m}$.
Now I have to prove that the infinite sum series $\sum_{m\geq 0}a_{m}(z)$ is well defined as a formal power series in the ring $\mathbb{C}[[z]]$.
I can't exactly figure out if this is what is meant since that would seem to imply that the coefficients in the above infinite sequence is a finite addition of coefficients from the summands $a_{m}(z)$. I think a little clarification of the above conditions would be helpful.
You have already got the point. We are allowed to write something like $$ \sum_{m\ge 0} a_m(z) \in \def\Pow{\mathbf C[\![z]\!]}\Pow \tag 1$$ although we have not topologized $\Pow$ yet (and hence convergence does not make sense) in a formal way. As $a_m(z) = \sum_{n\ge m} a_{m,n}z^n \in z^m\Pow$, each coefficient of the "sum" (1) is a finite sum of complex numbers, namely $$ \sum_{m\ge 0} a_m(z) = \sum_{m\ge 0}\sum_{n\ge m} a_{m,n} z^n = \sum_{n\ge 0} \left(\sum_{m=0}^n a_{m,n}\right) z^n \in \Pow $$ and the right hand term is well-defined.
One can of course give $\Pow$ a topology, such that the series in (1) converges, the so called $(z)$-adic topology, where a neighbourhood basis of $a(z) \in \Pow$ is given by $a(z) + z^m\Pow$, $m \ge 0$.