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The sum of an uncountable number of positive numbers

Suppose $f(x)>0$ for all real $x$, and $S$ is a set of uncountable many real numbers, how to prove that $\sum_{x\in S}f(x)=\infty$?

Alternately suppose $\sum_{x\in S}f(x)=k$, how to prove $|S|=N_0$ ?

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marked as duplicate by Asaf Karagila, Srivatsan, J. M., Matt N., Davide Giraudo Dec 25 '11 at 15:36

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hint: look at the cardinality of the set $\{x \in S| f(x) > 1/n\}$ and let $n\rightarrow \infty$ –  user20266 Dec 25 '11 at 15:25

2 Answers 2

Let $A_i=\{ x\in S : f(x)>1/i\}$ what is the cardinality of at least one of the sets $A_i$?

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if not, consider the sets $S_n=\{x\in S\ |\ f(x)>1/n\}$. then each of these sets must be finite and $S=\cup S_n$. this provides a contradiction to $S$ being uncountable.

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