1
$\begingroup$

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.

$\endgroup$
6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.