I found this problem from an online source.
I've just got two question
1) I think there is a typo in the solution, it should be $(x_n) \in \ell_1$ right?
2) I am guessing $c_0 \subsetneq \ell_\infty$ is already true because of the problem? Does the $\subset$ here actually mean "subset" or "subspace"? If they denote subspace, Ssouldn't the solution also show the three properties of a subspace or is that so bivous from the properties of sums and sequences (I can see it myself, for instance, $0$ belongs to all the sets) that you really don't need to show it?
FYI, I proved all of those myself already even though my proof is much longer and in more detail.