 Solution

For $1<p<\infty$, truncation makes the ${\ell }^p$ tail arbitrarily small, since ${\sum }_j|a_j{|}^p$ converges. Approximate the finite remaining coordinates by numbers in $\mathbb{Q}+i\mathbb{Q}$. The resulting countable set of finitely supported sequences is dense. ThereforeThe same construction also works for $p=1$, though that endpoint is not needed here.