Can we show , without considering the real numbers , that $\mathcal N$ is a maximal ideal of $\mathcal C$?

Question

Let $\mathcal C :=\{(r_n)\subseteq \mathbb Q : \forall k \in \mathbb Q^+ , \exists N_k \in \mathbb N : |r_n-r_m| < \dfrac1{k} , \forall n,m \ge N_k \}$ and $\mathcal N:=\{(r_n)\subseteq \mathbb Q : \forall k \in \mathbb Q^+ , \exists N_k \in \mathbb N : |r_n| < \dfrac1{k} , \forall n \ge N_k \}$ , it is known that $\mathcal C$ forms a commutative ring with unity under usual point-wise addition and multiplication of seuqneces ; with the knowledge of real numbers and archimedean property and all that , I can show that $\mathcal N$ is an ideal of $\mathcal C$ and $\mathcal C / \mathcal N \cong \mathbb R$ as rings , which shows that $\mathcal N$ is a maximal ideal of $\mathcal C$ . But I want to show , without considering the real numbers , that $\mathcal N$ is a maximal ideal of $\mathcal C$ so that I can use the definition $\mathbb R :=\mathcal C / \mathcal N$ , can this be done ?