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I am not sure how to go about this problem: Let $A$ be an uncountable set of positive real numbers. Show that there exists a finite subset of numbers from $A$ whose sum is larger than $2$.

By the Archimedean property, we get a positive integer $n$ such that $nx > 2$ for $x$ in $A$. But I am not sure how to get a finite subset from $A$ such that the elements' sum will be greater than $2$. Please help!

Kamal Saleh
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Emran Hossen
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  • There is no positive integer $n$ such that $nx>2$ for all $x \in (0,\infty)$ (it doesn’t work for $x=1/n$). – azif00 Jan 04 '23 at 21:36
  • See this question: https://math.stackexchange.com/q/20661/42969, and this comment: https://math.stackexchange.com/questions/20661/the-sum-of-an-uncountable-number-of-positive-numbers#comment44926_20661 – Martin R Jan 04 '23 at 21:37

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For each $n\ge 1$, consider the subset of elements of $A$ greater than $1/n$: $A_n = \{x\in A: x > 1/n\}$. Clearly $A$ is equal to the countable union $\bigcup_n A_n$. Since $A$ is uncountable by hypothesis, what can you conclude?

Alex Ortiz
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