Cardinality of an infinite set $S$ of real positive numbers such that the addition of the elements of any non empty finite subset is $\leq1$ I have to find the cardinality of an infinite set $S$ of real positive numbers such that the addition of the elements of any non empty finite subset of $S$ is $\leq1$.
We are working in ZFC. I have nothing but the obvious inequalities:
$\aleph_{0} \leq |S| \leq 2^{\aleph_{0}}$
Any help would be appreciated.
 A: Let $S$ be a set of positive reals such that the sum of finitely many elements from $S$ is always $\le 1$.
Take $n \in \mathbb{N}^+$. Then define $S_n = \{x \in S: x > \frac{1}{n} \}$.
Claim 1: $\cup_n S_n = S$. This is because each $x \in S$ is positive, so for each $x \in S$ there exists some $n$ such that $ 0 < \frac{1}{n} < x$, and this makes $x \in S_n$ for such an $n$.
Claim 2: $|S_n| \le n$. Or else, we'd have $n+1$ element of $S_n$ which would sum to at least $(n+1) \cdot \frac{1}{n} > 1$, contradicting the assumption on $S$.
Now conclude $S$ is at most countable. It can be infinite, just take any positive series with sum smaller than 1.
A: Suppose we have a set $S$ of positive real numbers.  Define the sum of the elements in $S$ to be the supremum over all sums of finite subsets of $S$:
$$\sum_{s\in S}s:=\sup\left\{\sum_{s\in S'}s\middle|S'\subset S,\;|S'|<\infty\right\}$$
Hint: With this definition, it can be shown that if $|S|$ is uncountable, then $\sum_{s\in S}s$ is infinite.
