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**armeros** I have a problem on this question.

Suppose $\displaystyle A \subseteq \mathbb{R}^+, b \in \mathbb{R}^+,$ and for every list $\displaystyle a_1, a_2, ..., a_n$ of finitely many distinct elements of $\displaystyle A, a_1+a_2+...+a_n \leq b$. Prove that $\displaystyle A$ is countable.

(Hint: For each positive integer $\displaystyle n$, let $\displaystyle A_n = \{x \in A | x \geq 1/n \}$. What can you say about the number of elements in $\displaystyle A_n$?)

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I can only conclude from a hint that $\displaystyle |A_1| \leq |A_2| \leq ... \leq |A_n|$ (where $\displaystyle |A_i|$ = the number of elements in $\displaystyle A_i$). It seems there is no link from this hint to the question. I need to show either there is a function $\displaystyle f:A \longrightarrow \mathbb{Z}^+$ that is one-to-one, or a function $\displaystyle g: \mathbb{Z}^+ \longrightarrow A$ that is onto.

If A = (0, 1), clearly this is uncountable. But I may choose *b* to be very large so that for any $\displaystyle n \in \mathbb{Z}^+, a_1+...+a_n \leq b$. Then, how could that A needs to be countable??