$I$ is some index set.
Let $l^1(I, \mathbb{R})$ denote the set of all sequences $a : I \to \mathbb{R}$ such that $\sum_{i \in I} |a(i)| < +\infty$.
Let $a_n \in l^1(I, \mathbb{R})$ for all $n \in \mathbb{N}$ such that:
$\sum_{i \in I}|a_n(i) - a_{n+1}(i)| < 2^{-n}$
After some working, this leads to
$\sum_{i \in I} \sum_{n = 0}^\infty |a_n(i) - a_{n+1}(i)| = 2 < +\infty$.
I want to show from here that $\lim_n a_n(i) = a(i)$ exists for all $i \in I$.
where $a \in l^1(I, \mathbb{R})$