Sum of uncountably many positive numbers is always infinite.

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Suppose a set $X=\{x_{\alpha }>0|\alpha \in I\}$ is an uncountable set. We will show that the sum of all elements of $X$ is infinite. Consider the family of the following sets.

 

$$S_n:=\{x_{\alpha }\in X\mid x_{\alpha }\ge \frac{1}{n}\}$$

 

Then

 

$$\bigcup _{n\in \mathbb{N}}{S}_n=X$$

 

$X$ is uncountable, so there exists $n_0\in \mathbb{N}$ such that $S_{n_0}$ is an infinite set. Since the sum of all elements of $S_{n_0}$ is infinite and $S_{n_0}\subseteq X$, the sum of all elements of X is also infinite. ■