Density of finitely supported sequences in l-p (${\overline{c_{00}}}^{\Vert \cdot {\Vert }_p}={\ell }^p$)

For $1\le p<\infty$, coordinate truncations converge in ${\ell }^p$, so $c_{00}$ is dense. It is not dense in ${\ell }^{\infty }$: every finitely supported sequence is at distance at least one from the constant-one sequence.