Consider the normed space $\ell_p$ for $p\geq 1$ of infinity sequences {$x_k$} s.t. $\sum_{i=1}^{\infty}\left|x_i\right|^p < \infty$ with norm defined as;
$\left\|x\right\|_p=\left(\sum_{i=1}^{n}\left|x_i\right|^p\right)^{1/p}$
The question contains two parts, for the first part I have been able to complete by proving that $\left\|x\right\|_p$ is indeed a norm by applying Minkowski's Inequality for the triangle inequality which follows from duality using Holder's inequality (which follows from Young's inequality).
But now I have been asked to show that this normed space $\ell_p$ is also separable which I am stumped on. I know that the space contains infinity sequences {$x_k$} but why must every nonempty open subset of the space $\ell_p$ contains at least one element of the sequence?
Any help or hints greatly appreciated.