Prove $c_0$ is a banach space.

Question

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.

Answer 1

Yes, you're looking at a sequence of sequences. The “distance” between two bounded sequences $(x_1,x_2,\ldots)$ and $(y_1,y_2,\ldots)$ is given by the so-called $\ell^{\infty}$-norm, which is defined as $$\|(x_1,x_2,\ldots)-(y_1,y_2,\ldots)\|_{\infty}\equiv\sup_{n\in\mathbb N}|x_n-y_n|.$$ The Cauchy property and convergence of a sequence of sequences should be evaluated with respect to this norm.

Answer 2

$c_0$ is a vector subspace of the closed space $c$, so you just have to prove:

If $x_n \rightarrow x$ where $x_n \in c_0$ and $x\in c$, then $x\in c_0$.