Convergent Generalized Sum of Positive Reals has Countably Many Non-Zero Terms

Theorem
Let $\left({a_i}\right)_{i \in I}$ be an $I$-indexed family of positive real numbers.

That is, let $a_i \in \R_{>0}$ for all $i \in I$.

Suppose that $\displaystyle \sum \left\{{a_i : i \in I}\right\}$ converges.

Then the set $I_{>0} := \left\{{i \in I: a_i > 0}\right\}$ is countable.

Proof
Denote, for $\lambda \in \R$, $I_{>\lambda} := \left\{{i \in I: a_i > \lambda }\right\}$.

Then as $\displaystyle \sum \left\{{a_i : i \in I}\right\}$ converges, necessarily all of the sets $I_{> \frac 1 n}$ are finite.

However, we have that $\displaystyle I_{>0} = \bigcup_{n=1}^\infty I_{> \frac 1 n}$.

From Union of Countable Sets, it follows that $I_{>0}$ is countable.