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.
Best Answer
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.