show that sum of $f(x)$ which $x$ is in $X$ is same to sup{sum of $f(x)$ which $x$ is in $F$ is finite subset of $X$}

Question

in this question Exercise on measure theory

how can we say that $$\sum_{x\in X}f(x) = \sup\left\{\sum_{x∈F}f(x),\ F\subseteq X,\ F\ \text{finite} \right \}$$?

left side is sum of all element in $X$ and right side is at most $\sup\{\text{only small part of}\ X\}$

show that this equation.

Answer 1

Patrick is right. For instance, suppose that $X$ is the set of integers and $f(x)=1/(x^2+1)$. Then

$$\sum_{x=-\infty}^\infty \frac{1}{x^2+1} = \sup_n \sum_{x=-n}^n \frac{1}{x^2+1}.$$