2
$\begingroup$

Possible Duplicate:
The sum of an uncountable number of positive numbers

Suppose $f(x)>0$ for all real $x$, and $S$ is a set of uncountable many real numbers, how to prove that $\sum_{x\in S}f(x)=\infty$?

Alternately suppose $\sum_{x\in S}f(x)=k$, how to prove $|S|=N_0$ ?

  • 1
    hint: look at the cardinality of the set $\{x \in S| f(x) > 1/n\}$ and let $n\rightarrow \infty$2011-12-25

2 Answers 2