= Solution
Take
$$
\boxed{\psi_1(\alpha)=\alpha_1.}
$$
It is evaluation against the first coordinate vector of $c_0$, hence continuous for $\sigma(\ell^1,c_0)$. The preceding continuous-dual result says that a functional continuous for $\sigma(\ell^1,c)$ must have coefficient sequence $(\gamma_\infty,\gamma_1,\gamma_2,\ldots)$ for a convergent sequence $\gamma$. To represent $\alpha_1$, it would require $\gamma_\infty=1$ but every $\gamma_j=0$, impossible.
Directly, $e_1-e_n\to0$ in $\sigma(\ell^1,c)$ because its pairing is $\gamma_\infty-\gamma_{n-1}\to0$, whereas $\psi_1(e_1-e_n)=1$ for $n\geq2$.
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