What's wrong in this reasoning of $l_\infty$ separability?

Question

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in it. This "proof" is given below, any idea of what's wrong?

1. $c_o^* = l_1$
2. $c_0 \subset l_\infty$
3. (follows from 2) $l_\infty^* \subseteq c_0^*$
4. (follows from 1 and 3) $l_\infty^* \subseteq l_1$
5. $l_1$ is separable
6. (follows from 4 and 5, as both $l_\infty^*$ and $l_1$ are metric spaces) $l_\infty^*$ is separable
7. (follows from 6, as $l_\infty$ is normed space) $l_\infty$ is separable

Answer 1

From (2) one can only deduce that there is a map $q:\ell_\infty^* \to c_0^*$. You don't know that the map is one-one. And actually it is onto. Using this, you can show that $c_0^*$ is a complemented subspace of $\ell_\infty^*$.

Answer 2

$c_0$ is a closed subspace of $\ell_\infty$, and by virtue of Hahn-Banach, every element of $(c_0)^*$ extends to an element of $(\ell_\infty)^*$, and it does so in infinitely many ways. In particularly, there is (using Zorn's Lemma) an injection $$ j: (c_0)^*\to (\ell_\infty)^*. $$