  Given $1< p < q < \infty$ it is well know that the natural embedding of
Lebesgue sequence spaces $\ell_p \hookrightarrow \ell_q$ is strictly singular.
In this paper we extend this classical results and show that even the natural
non-compact embedding between Lebesgue and weak Lebesgue sequence spaces
$\ell_{p}\hookrightarrow \ell_{p,\infty}$ is strictly singular.
