Understanding weakly dense.
Solution 1:
Here is a sketch as to how you can proceed:
Claim: If $C$ is a convex subset of a normed space, then the weak closure and the norm closure of $C$ are equal.
Proof: Apply the Hahn-Banach separation theorem. Details are left for you. $\quad \square$
Corollary: The weak closure of $c_0$ is equal to the norm closure of $c_0$ in $\ell^\infty$. Since $c_0$ is norm-closed in $\ell^\infty$, we deduce that $c_0$ is weakly closed in $\ell^\infty$ as well. So basically you are asked to show that the inclusion $c_0 \subseteq \ell^\infty$ is strict, which is trivial (consider the function $n \mapsto 1$).