If a sequence of summable sequences converges to a sequence, then that sequence is summable. Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge).  Then if they have a limit, the limit sequence $(b_i)$ is also summable.  All under the sup norm for sequences.
Let $(a_i)^n$ sum to $c_n$.  I.e. for all $\epsilon \gt 0$, there's $N$ such that $m \gt N \implies |\sum_{i=1}^m a_i^n - c_n| \lt \epsilon$.  I want to show that there's $b$ such that $|\sum_{i=1}^m b_i - b| \lt \epsilon$ similarly.  Let $b = \lim c_n$.   Where to?

Let $(a_i)^n$ be a sequence of summable sequences that converges to $(b_i)$, and let each be summable to $c_n$ respectively. I want to show that the sequence $c_n$ is Cauchy and thus has a limit $c \in \mathbb{C}$.  $|c_n - c_m| \approx \lim_{N\rightarrow \infty}|\sum_{i=1}^{N} a_i^n - \sum_{i=1}^{N} a_i^m|$.  Still working...
