I want to show that $l^p$ is separable.
I got the hint to use the set $E:=\{e_n: n \in \mathbb{N}\}$, where $e_n$ denotes the sequence with a 1 in the n-th position and else zeros.
There is a corollary that stats that, if there exists a countable subset $A$ in a normed vector space $X$ such that $\overline{span A}=X$, then $X$ is separable.
Thus I want to show that $\overline{span E}=l^p$.
Since $(l^p,\lVert.\rVert_p)$ is a metric space (the norm induces the metric), the closure of $span E$ are all limits of sequences with elements in $span E$.
This means that for some $x=(x_n)$ in $l^p$, I have to construct a sequence $y_n=(y_{1,n},y_{2,n},...)$ such that $y_{i,n} \in spanE$ and $y_n \rightarrow x$.
But I don't know how to construct/find this sequence. (I do not even know if my approach is correct).