Show that the subspace $c_{0} = \{ x = (x_{1}, x_{2}, \ldots ) \in \ell^{\infty} : \lim x_n = 0 \}$ is a maximal ideal in $c = \{ x = (x_{1}, x_{2}, \ldots ) \in \ell^{\infty} : \lim x_n \text{ exists } \}$ by finding a linear multiplicative functional with $c_{0}$ as its kernel.
I already showed that $c_{0}$ is a proper, closed ideal. I am stuck on showing that it is a maximal ideal by finding a multiplicative linear functional $\phi$ with $\ker(\phi) = c_{0}$.
Consider $\phi(x_1,x_2,\dots):=\lim_n x_n$.
It's clearly linear and maps to the scalar field, its kernel is just $c_0$, and thus it's a one codimensional subspace, hence must be maximal.