I have the following definition of separable: We say that a space X is separable if there exists a countable subset S such that $\bar S=X$.
Using this I want to prove that $l^p$ is separable. Would it work to take $S=\{(x_n)_n: x_n\in \Bbb{Q}: \sum_n|x_n|^p<\infty\}$? I only want to know if I'm on the right track or if I'm totally wrong. So my idea is to use that $\Bbb{Q}$ is countable, that's why I get my S.