For $x=(x_n)\in\ell^p$ with $1\leq p<\infty$, let
$$
x^{(N)}=(x_1,\ldots,x_N,0,0,\ldots)\in F.
$$
Then
$$
\lVert x-x^{(N)}\rVert_p^p
=\sum_{n>N}|x_n|^p\longrightarrow0
$$
because the defining nonnegative series converges. Thus $F=c_{00}$ is dense in the <l-p sequence space> for every finite $p$; this is the <density of finitely supported sequences in l-p>.

It is not dense in the <l-infinity sequence space>. For the constant sequence $\mathbf1=(1,1,\ldots)$ and any finitely supported $y$, some tail of $y$ is zero, so
$$
\lVert\mathbf1-y\rVert_\infty=1.
$$
\b[Hence $\mathbf1$ does not belong to the closure of $F$.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 22H / a / Solution

{title2=$\ell^\infty$}
{wiki=Sequence_space#ℓp_spaces}

The space $\ell^\infty$ consists of bounded scalar sequences with norm $\lVert x\rVert_\infty=\sup_n|x_n|$.


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