TITLE: Question from "An introduction to measure theory" by Terence Tao
QUESTION [6 upvotes]: If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} < \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is uncountable.

REPLY [5 votes]: Consider the set of $\alpha$ for which $x_\alpha > 1/n$, for each natural $n$.  It must be finite, and there are countably many integers, so all but countably many $x_\alpha$ are less than each $1/n$, and thus zero.