Let $A$ be a family of positive real numbers, defined $\sum A = \sup\left\{ \sum F:F\subseteq A,F \text{ is finite} \right\} $. Show that $\sum A <+\infty $ only if $A$ is countable.

Doubt: Let $B\_n =\left\{ x\in A:\frac 1 { n+1 } \le x < \frac 1 n \right\} $ .
Also
$$B\_0 =\left\{ x\in A:x\ge 1 \right\} $$ and $\bigcup\_{k=0}^{+\infty} B\_k =A$ if $A$ is uncountable $\exists n\_0 : B\_{n\_0}$ is finite.

If $x\in B\_{n\_0} \Rightarrow x\ge \frac 1 {n\_0} +1$ then $\sum A >\sum \left( \frac 1 { n\_{n\_0}+1 } \right) =+\infty$