I am just a beginner. So please be patient with me.
Consider $X=\{\frac 1 n : n \in \mathbb N\} \cup \{0\}$. Prove that the vector space of continuous functions on $X$ is linearly isomorphic to the space of convergent sequences in $\mathbb R$. Using this result, conclude that the set of convergent sequences in the NLS of all bounded real sequences under the sup norm is complete.