Prove that a set is countable Let $A\subseteq \Bbb R$, A is an infinite set of positive numbers.
Suppose, $\exists k \in 
\Bbb Z$ for all finite $B\subseteq A$.
$\sum_{b\in B}b<k$
Prove $A$ is countable. 
I have a hint: $A_n=\lbrace a \in A | a>\frac1n\rbrace$
I understand that $B \in \Bbb Q$ and $k$ can be the last element of $A$ but I'm not really sure if that's right or what to do with it.
Any help would be appreciated.

HINT: Prove that each $A_n$ must be finite, and hence that $A=\bigcup_{n\in\Bbb Z^+}A_n$ is countable. If $A_n$ is infinite, and all of its elements are bigger than $\frac1n$, then there is a finite $B\subseteq A_n$ such that ... ?
