Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$

$$ \sum_{i}a_iz_i^{n}=0 $$ does this imply that $a_i=0$ for any $i$?

Edit: For the "finite dimensional" case, when we have $n$ *distinct* $(z_i)$, then $(a_i)$ must be $0$. This amounts to solving a homogeneous system of $n$ equations with $n$ unknowns, which only has the trivial solution in the case of distinct $(z_i)$. I am really curious what happens in the infinite dimensional case. My intuition tells me the same must be true, but I don't have a proof for it.

Edit 2: Very interesting, looking were this question originated, the fact that all $a_i=0$ when $(a_i)\in l_1$ is "expected". I was hoping to get a counterexample otherwise. However, if a non-trivial sequence $a_i$ exists (at least for some sequences $z_i$), I would "expect" to be able to choose it in $l_2$. Looking at Davide and Julien answers below, it seems $(a_i)\in l_1$ is an essential assumption in their argument.