Question regarding a step in Wikipedia's proof of the Radon-Nikodym Theorem

Question

I have a question regarding Wikipedia's proof of the Radon-Nikodym theorem for finite measures:

Why does there exist a sequence of functions $\{f_n\}$ in $F$ such that $$\lim_{n\to\infty}\int f_n\ d\mu = \sup_{f\in F}\int f\ d\mu?$$

Answer 1

It is just the definition of $\sup$. It can be approximated arbitrarily well taking appropriate elements of $F$. If it is $<+\infty$, we can take $f_n \in F$ s.t. $$\int f_n d\mu > \sup_{f\in F} \int f d\mu-\frac{1}{n}$$

while if it is $+\infty$ we can take $f_n \in F$ s.t. $$\int f_n d\mu >n$$