Let $I$ be a set $(a_i)_{i\in I}$ be a sequence with positive entries and let $\sup\left\{\sum_{i\in F} a_i \mid F\subseteq I, F \ \text{finite}\right\}<\infty.$ I want to show that in that case the set $J$ of indexes that index numbers that aren't $0$ is at most countably infinite.
Consider this Wikipedia proof: It consists of writing $J$ as $J=\bigcup_{n} A_n$, where $A_n:=\{ i\in J \mid \frac{1}{n}
The problem I have is with the first inequality: $\frac{1}{n} |A_n| \leq \sup\{\sum_{i\in F} a_i \mid F\subseteq I, F \ \text{finite}\}$. At first I thought "it's obvious", since all elements in $A_n$ are greater than $\frac{1}{n}$ so one can bound every $a_j\in A_n$ from below with $\frac{1}{n}$ - but the problem is more subtle I think: Namely at this stage of the proof, we can't exclude, that $A_n$ is not countably infinite; and in that case the sum I can't see any way to prove that inequality!