Let $a_i, i \in \mathbb{R}$ be non-negative real numbers such that $\sup\left\{\sum_{i \in F} a_i \mid F \subseteq \mathbb{R} \text{ a finite subset}\right\}$ is finite. Show that $a_i = 0$ except for countably many $i \in \mathbb{R}$. Give an example to show that 'countably' cannot be replaced by 'finite'. (Hint: consider $F_n := \left\{i \left\lvert\, a_i \geq \frac{1}{n}\right.\right\}$.)