Consider a set $B$ of positive real numbers such that the sum of elements in any finite subset of $B$ is always less than or equal to $2$. Show that $B$ is countable.
I'm trying to find a bijection between $\mathbb{N}$ and $B$ but it's not clear how I would do this. I have a feeling I should be using the fact that if you have finite subsets $S$ and $T$, the sum of elements in $S \cup T$ is also $ \leq 2$… but I don't see that going anywhere.
I'd appreciate a hint, with a full solution in spoiler markdown if you can manage it.
Best Answer
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.