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Post Closed as "too localized" by Andres Caicedo, Andreas Thom, Felipe Voloch, S. Carnahan
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# Sums of uncountably many real numbers

Suppose $S$ is an uncountable set, and $f$ is a function from $S$ to the positive real numbers. Define the sum of $f$ over $S$ to be the supremum of $\sum_{x \in N} f(x)$ as $N$ ranges over all countable subsets of $S$. Is it possible to choose $S$ and $f$ so that the sum is finite? If so, please exhibit such $S$ and $f$.