Series of complex numbers over an uncountable set 
Question: Let I be an arbitary index set and $(a_i)_{i\in I} \subset \Bbb C$ such that $\sum_{i \in I}a_i$ converges. Show that there exists a countable set $I_0 \subset I$ such that $a_i=0$ for all $i \in I\setminus I_0$.  

I think I am able to prove this if all the real and imaginary parts of $a_i$ are positive but for any sequence $a_i$ I'm struggling. I feel like I want to use the Cauchy criterion, namely that since we have convergence, we also have that for all $\epsilon>0$ there exists a set $J_{\epsilon}$ such that $$\bigg|\sum_{i\in J\setminus J_{\epsilon}}a_i\bigg|<\epsilon$$ for all $J\supset J_{\epsilon}$ where $J\subset I$ is finite. We could then choose $\epsilon=\frac{1}{n}$ to generate a family of $J_n$ and possibly take the union over $n$ to create the countable set I'm after. But beyond that I'm stuck. 
Any help is much appreciated.

You do not say what is your definition of the convergence of $\sum_{i \in I} a_i$, but I guess it is this:
$\sum_{i \in I} a_i = a$ if for each $\epsilon >0$ there exists a finite $I_\epsilon \subset I$ such that for all finite $J \subset I$ which contain $I_\epsilon$ one has $\lvert \sum_{i \in J} a_i - a \rvert < \epsilon$.
This implies that for all $j \notin I_\epsilon$
$$\lvert a_j \rvert = \lvert \sum_{i \in I_\epsilon \cup \{j \}} a_i - \sum_{i \in I_\epsilon } a_i \rvert = \lvert \sum_{i \in I_\epsilon \cup \{j \}} a_i - a + a - \sum_{i \in I_\epsilon } a_i \rvert \\ \le \lvert \sum_{i \in I_\epsilon \cup \{j \}} a_i - a \rvert + \lvert a - \sum_{i \in I_\epsilon } a_i \rvert  < 2\epsilon .$$
Let $J_n = \{ i \in I \mid \lvert a_i \rvert \ge 1/n \}$ and $I_0 = \{ i \in I \mid a_i \ne 0 \}$. We have $I_0 = \bigcup_{n=1}^\infty J_n$.
Assume that some $J_n$ is infinite. For $\epsilon = 1/2n$ choose $I_\epsilon$ as above. Pick any $j \in J_n \setminus I_\epsilon$. Then $\lvert a_j \rvert < 2 \cdot 1/2n = 1/n$ which is a contradiction.
Hence all $J_n$ are finite and $I_0$ is countable.
Note that this proof can be generalized to arbitrary normed linear spaces.
