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.)

Finally, observe that $S = \bigcup_{n=1}^\infty A_n$. Since each $A_n$ is finite, $S$ is at most countably infinite, which contradicts to the given assumption that $S$ is uncountable.