Question. I'm familiar with the parallel of this in regards to function spaces ($C_c(\mathbb{R})$ is dense in $L^p$), but how does one show this for sequence spaces?
We define $c_{00}$ as
$$c_{00} = \left\{x = (x_k)_{k \in \mathbb{N}} :~\text{finitely many}~x_k \neq 0\right\}.$$
What I have. Ideally, I think the best route is to show convergence in $\ell^p$-norm. That is, for any $x \in \ell^p$, show that there exists a $x_n \in c_{00}$ such that $x_n \to x$ in $\ell^p$-norm.
So, we want to show
$$\lim_{n \to \infty} ||x - x_n||_p = \dots = 0. \tag{1}$$
But, I'm drawing a blank on inequalities to use here?
The inequality that you need to use here is that if $\sum_{n\geqslant 1}|a_n|^p<\infty $ then for each $\epsilon >0$ there is some some $N\in \mathbb N $ such that $\sum_{k\geqslant n}|a_k|^p<\epsilon $ for all $n>N$, therefore for $s:=(x_1,x_2,\ldots )$ setting
$$ s_n:=(x_1,x_2,\ldots ,x_n,0,0,0,\ldots ) $$
you find that
$$ \|s-s_n\|_p^p\leqslant \sum_{k\geqslant n}|x_k|^p<\epsilon $$