I was checking a proof in another question: Show that the sum of a non-negative series is the supremum of its finite partial sums
And just at the question statement:
$$\sum _{n=0}^{\infty}x_{n}= sup\left \{\sum _{n\in F}x_{n}:F\subset \mathbb{N} \hspace{0.1cm} is finite \right \}$$
I don´t get what the elements of the set on the right side are. I know F is an index set, and i also know that the elements to be summed are non negative real numbers. What confused me is whether the numbers of elements being summed changes or is the length of the index set F what changes?
So, can someone help me understand what the elements of the set from where the supremum is going to be gotten are?