How to prove that given an uncountable subset $A$ of $R^+$, there exits a finite number of elements in A whose sum is arbitrarily large?
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Does $R^+$ denote the positive reals? – Abel Apr 12 '13 at 17:41
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yes. positive reals. – GA316 Apr 12 '13 at 17:42
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HINT: Let $I_0=(1,\to)$, and for $n\in\Bbb Z^+$ let $I_n=\left(\frac1{n+1},\frac1n\right]$. $\{I_n:n\in\Bbb N\}$ is countable, and $A$ is uncountable, so there must be some $n_0\in\Bbb N$ such that $A\cap I_{n_0}$ is uncountable. Show that $A\cap I_{n_0}$ has finite subsets with arbitrarily large sums.
Brian M. Scott
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sorry. I dont understand. Why $A \cap I_{n_{0}}$ should have finite finite sunsets with arbitrary cardinality? – GA316 Apr 12 '13 at 17:55
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2@GArunkumar: Because it’s infinite, and every element of it is greater than $\frac1{n+1}$. If you take a subset $F$ of $A\cap I_{n_0}$ that contains $k$ elements, and each of those $k$ elements is greater than $\frac1{n+1}$, then the sum of the elements of $F$ is greater than ... what? – Brian M. Scott Apr 12 '13 at 17:58
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1ya. by choose large k, we can make $k\frac{1}{n+1}$ as large as possible.thanks. – GA316 Apr 15 '13 at 13:19
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Let $A_n = \{x \in A : x > \frac 1n\}$. Then at least one set $A_n$ is infinite because...
Umberto P.
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one $A_n$ will be infinite. but from this what is the conclusion. can you please explain me more clearly – GA316 Apr 15 '13 at 13:15
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thanks. I understand your point. there exist finite subset $A_n$ which sum can be made large. – GA316 Apr 15 '13 at 13:20
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How does the conclusion that there exists a finite subset whose sum can be made large follow from showing that at least one $A_n$ is infinite? – gpickart Feb 16 '18 at 06:23