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.


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.

  • $\begingroup$ Cute, I like it. Did you think of this just for this question or have you answered similar questions (so you had this arrow ready to shoot)? $\endgroup$ – marty cohen Sep 27 '17 at 2:00
  • 1
    $\begingroup$ @martycohen Nice metaphor! Yes, I had it ready. :-) $\endgroup$ – José Carlos Santos Sep 27 '17 at 6:15

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