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

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Theorem

Let $\family {a_i}_{i \mathop \in I}$ 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 $\ds \sum \set {a_i: i \in I}$ converges.

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

Proof

For $\lambda \in \R$, let $I_{>\lambda} := \set {i \in I: a_i > \lambda}$.

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

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

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

$\blacksquare$