Sum of a set analysis 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 $\sum A >\sum \left( \frac 1 { n_{n_0}+1 } \right) =+\infty$
 A: 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 infinite.
If $x\in { B }_{ { n }_{ 0 } }\Rightarrow x\ge \frac { 1 }{ { n }_{ 0 }+1 } $ then $\sum { A } >\sum { \left( \frac { 1 }{ { n }_{ 0 }+1 }  \right) =+\infty  } $
A: If this is an exercise in a course you're taking, the proof should be written in a way that displays your understanding of why it is correct, rather than in a way that from some point of view is correct.
And if it's not an exercise on which you're being graded, it should be written in a way that that makes it clear to the reader, rather than in a way that from some point of view is correct.
I mention this because this appeared:
$$
\text{if $A$ is uncountable $\exists n_0 : B_{n_0}$ is finite.}
$$
As someone has already pointed out, that should say "infinite" rather than "finite", but I would add that even if it had correctly said "infinite", it should be explained further.
If every one of the sets $B_1,B_2,B_3,\ldots$ is finite $B_1 \cup B_2\cup B_3 \cup \cdots$ is countable, since one can enumerate it by first listing the members of $B_1$, then those of $B_2$, and so on. I'd write that out explicitly.
