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

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Theorem

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

That is, let $a_i \in \R_{\ge 0}$ 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 $\displaystyle \sum \left\{ {a_i: i \mathop \in I}\right\}$ converges, necessarily all of the sets $I_{> \frac 1 n}$ are finite.

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

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

$\blacksquare$