# Showing a set is finite or countable

Let $B$ be a set of positive real numbers with the property that adding together any finite subset of elements of $B$ always gives a sum of 2 or less. Show $B$ must be finite or countable.

I do not know where to start with this proof. Any help is appreciated.

• Can we perhaps get a better title? – Asaf Karagila Dec 4 '17 at 7:52
• – Akash Gaur Dec 5 '17 at 9:48

For each $n\in\mathbb N$, let$$B_n=\left\{b\in B\,\middle|\,b\geqslant\frac2n\right\}\subset B.$$Of course, $B_n$ can have no more than $n-1$ distinct elements; otherwise, the sum of $n$ distinct elements of $B_n$ would be grater than $2$.
But$$B=\bigcup_{n\in\mathbb N}B_n.$$Since $\mathbb N$ is countable and each $B_n$ is finite, $B$ is countable.