Identify the boundary $\partial c_{00}$ in $\ell^p$, for each $p\in[1,\infty]$. Also, for each $p\in[1,\infty]$, identify the completion of the metric space $(c_{00},d_p)$.
Note that $c_0$ is the set of all real sequences that converge to $0$, and note that $c_{00}:= \left\{x=\{x_n\}_{n=1}^\infty\in c_0\,:\,\text{ there is an $N\in\mathbb{N}$ such that $x_n=0$ for all $n\geq N$}\right\}$
If $p<\infty$, then $\overline{c_{00}}=\ell^p$ and $\mathring{c_{00}}=\emptyset$. Therefore, $\partial c_{00}=\ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(\ell^p,d_p)$.
In $(\ell^\infty,d_\infty)$, it is still true that $\mathring{c_{00}}=\emptyset$, but now $\overline{c_{00}}=c_0$. So, $\partial c_{00}=c_0$ and the completion of $(c_{00},d_\infty)$ can be identified with $(c_0,d_\infty)$.