In other words: Suppose that there is uncountable subset $X$ of the positive Real numers. Prove that there is countable subset $S$ of $X$ that the sum of elements in $S$ is infinite.
The only thing I thought about was the Natural number but they don't have to be a subset of $X$ if $X$ is only real number without natural number.
For $n \in \Bbb N$, consider the sets $A_n := [\frac 1 n , \frac 1 {n-1})$, with $\frac 1 0 := \infty$. Then suppose $A_n \cap X$ is finite for every $n\in\Bbb N$. Observe that $\bigcup_{n=1}^\infty A_n = \Bbb R _{>0}$. So, $$X = \bigcup_{n=1}^\infty A_n\cap X$$ would be an countable set as countable union of finite sets. Contradiction. Thus there exists an $m\in \Bbb N$ such that $A_m\cap X$ is infinite. Now take some infinite countable subset $S$ of $A_m\cap X$, for example take successively different elements of $A_m\cap X$. Obviously $S\subseteq X$, and since for every $s\in S $ holds $s > \frac 1 m > 0$ $$\sum_{s\in S} s \geq \sum_{s\in S} \frac 1 m = \infty$$