Countability of set of positive reals with bounded sum for all finite subsets

Solution 1:

Hint: Let $B_0$ be the set of elements of $B$ that are greater than $1$. For every positive integer $n$, let $B_n$ be the set of elements of $B$ that are in the interval $\left[\frac{1}{n},\frac{1}{n+1}\right)$.

The set $B$ has been decomposed into a countable union of finite sets.

Solution 2:

You can prove it by establishing the following fact:

Let $M$ is an indexing set and for all $j \in M$, let $a_j \in [0,\infty[$, define $$\sum_{j \in M} a_j = \sup\left\{\sum_{j \in N} a_j\mid N \subseteq M, |N| < \aleph_0\right\}$$ Then $\sum_{j \in M} a_j < \infty$ only if only a countable number of $a_j$s is non-zero.

Hint: Consider sets of the form $S_n = \{a_j\mid a_j \geq \frac{1}{n}\}$.

Or, you can convert this to integration w.r.t. counting measure on $B$. Then it immediately follows from a property about integration.