A countability problem of $\mathbb{R}$ I am having a really hard time coming up with a proof for this problem. For every finite set $F\subseteq \mathbb{R}$, let $\Sigma(F)$ denote the sum of the numbers in $F.(\Sigma(\emptyset)=0)$. Show that if $S\subseteq (0,+\infty)$ and $\Sigma(F)\leq100$ for every finite set $F \subseteq S$, then $S$ is finite or countable. I can't see why $S$ could be countable but not uncountable. Any suggestions are much appreciated.
Hint: Let $S_n=\{s\in S\mid s&gt;\frac{1}{n}\}$. How many elements can be in $S_n$ if $\sum(F)&lt;100$ for all finite $F\subseteq S$?
$S=\{1,1/2,1/4,1/8,1/16,\ldots\}$ is an example of a countably infinite set that meets the requirements.