Let $S \subset(0,1)$ be an uncountable set and let

$$

T=\left\{\sum_{a \in A} a: A \subset S, \# A<\infty\right\}

$$

be the set of sums of elements in finite subsets of $S$. Prove that $\sup T=\infty$. (Thus, unlike in the countable case, an infinite sum of elements in an uncountable set cannot be bounded).

I think if I prove that there exists some $\varepsilon>0$ such that $S \cap(\varepsilon, 1)$ is infinite, this problem gets solved, but I can't see how the intersection of two bounded sets is going to be infinite!

## Best Answer

Hint:If for every $n\in\mathbb{N}$ the set $S\cap (\frac{1}{n}, 1)$ was finite then $S$ would be countable.