Sum of uncountably many positive numbers is always infinite.

pixabay

 

 

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.

 

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

 

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. ■