Uncountable summation diverges
Problem If $S$ is an uncountable collection of positive real numbers, then the summation of $S$ diverges. Proof Suppose otherwise, the summation converges. For $n = 1, 2, …$, denote $A_n = \{x \in S \mid x \geq \frac{1}{n}\}$. Notice that $A_n$ is a subset of $S$, so its summation must converge. Since each element of $A_n$ has a lower bound $1/n > 0$, the cardinality of $A_n$ must be finite. (Otherwise $\sum A_n$ diverges.) ...