The subspace of null sequences $c_0$ consists of all sequences whose limit is zero. Prove that $c_0$ is a closed subspace of $C$ (The space of convergent sequences), and so again a Banach space.
There's something I don't understand. I know we have to prove that every Cauchy sequence on $c_0$ is convergent on $C$ in order to prove $c_0$ is closed on $C$. But, that Cauchy sequence will be a sequence of sequences? Because the elements of $C$ and $c_0$ are sequences. I'm really confused.