find a continuous functional $f \in (\ell^{1})^{*}$ so that $M \subseteq \ker f$
Note $M:=\{x \in \ell^{c}: \sum\limits_{n \geq 1}x_{n}=0\}$ and
$\ell^{c}=\{x \in \ell^{\infty}:x_{n}=0 \operatorname{for all but finitely many} n \in \mathbb N\}$
My only idea:
setting $f(x)=\sum\limits_{n \geq 1}x_{n}$ and we know for $x \in M$ that $\sum\limits_{n \geq 1}x_{n}=0$ thus $M\subseteq \ker f$
Is this correct?
Why does this suffice to show that $M$ is not dense in $\ell^{1}$
What you have done is correct. For the last part note the kernel of any continuous linear functional is closed. Hence, if $M$ is dense then it must be equal to $\ell^{1}$. This is certainly false, right?