Proof that if unordered sum is finite, then there are finitely many positive terms

Question

I'm confused about a statement I've seen.

For $\{c_{i} : i \in I\}$, and $P = \{i \in I : c_{i} > 0\}$ it states:

$\sum_{i \in I} c_{i} < \infty \implies $P countable

I understand this is true intuitively but I don't know how you'd formulate the proof for it.