Maximal ideal of Cauchy nullsequences

Question

Let $(K, \left|\phantom{x}\right|)$ be a valued field. Let $\mathscr{C}$ be the commutative ring with unity given by Cauchy sequences of elements in $K$, with $+$ and $\cdot$ defined term by term. Finally, let $\mathscr{M}$ be the subset of $\mathscr{C}$ given by Cauchy sequences converging to $0$. I've proved that $\mathscr{M}$ is an ideal of $\mathscr{C}$. Now I want to prove that is maximal. To do this, I think it suffices to show that $\mathscr{C}-\mathscr{M}\subseteq\mathscr{C}^{\star}$, i.e. that every $\{a_n\}$ in $\mathscr{C}$, not in $\mathscr{M}$, is invertible. I've been able, given such a sequence $\{a_n\}$, to find a sequence $\{b_n\}$ in $\mathscr{C}$, such that $\{a_n\}\cdot\{b_n\}=\{1\}+\zeta$, where $\zeta$ is a sequence in $\mathscr{M }$. How can I conclude, from this, that $\{a_n\}$ is invertible?