Let
- $\omega$ be the space of all real sequences $(x_{n})_{n}$
- $c$ be the space of all convergent real sequences
- $c_0$ be the space of all real sequences converging to $0$
- $l_2 = \{(x_{n})_{n}\in{\omega}\lvert \sum_{n=1}^{\infty} |x_{n}|^2 \lt\infty \}$
Is the set $l_2$ everywhere dense in the space $c_0$, in the space $c$, and in the space $\omega$?
First of all, the question does not specify any metric, so I take the metric in both spaces $c$ and $c_0$ as the usual metric $d(x_n,y_n)=\sup_{n\in{\Bbb{N}}}\lvert {x_n-y_n}\rvert $.
Now, if $(x^n_{k})_{n}$ is a sequence in $l_2$ (that is, for each $n\in{\Bbb{N}}$, $ (x^n_{k})_{k}\in{\omega} $) converging to $(x_{k})_{k}\in{\omega}$, then each $(x^n_{k})_{k}\in{\omega}$ converges to $x_{k}\in{\Bbb{R}}$. But I think this does not guarantee that the real sequence $(x_{k})$ converges to $0$, or converges at all. So, it is enough to find an example of a sequence $(x^n_{k})_{n}$ in $l_2$ converging to a divergent sequence $(x_{k})$. Any idea?
Then, I believe this also should imply that the closure of $l_2$ is exactly $\omega$ meaning that $l_2$ is everywhere dense in $\omega$. Here, I will try to use the Fréchet metric $d(x_n,y_n)=\sum_{n=1}^{\infty} 2^{-n} \frac {|x_{n}-y_n|}{1+|x_{n}-y_n|} $.
Is my intuition correct, how can I find such an example, and how can I prove the last statement? I had a very rough time tackling with this question so any idea is highly appreciated.
If $(x_n) \in c_0$ the $(x_1,x_2,..,x_n,0,0,...)$ is sequence in $\ell^{2}$ which converges to $(x_n)$, so $\ell^{2}$ is dense in $c_0$.
The sequence $(x_n)=(1,1,1,...) $ is in $c$ but there is no element $(y_n)$ of $\ell^{2}$ with $\sum |x_n-y_n|^{2} <\frac 1 4$: Because $|x_n-y_n| <\frac 1 2$ and $y _n \to 0$ so we get $1=|x_n| < 1$ for $n$ sufficiently large. So $\ell^{2}$ is not dense in $c$.
I will give a hint for $\omega$: Use the idea used for $c_0$; Note that $\sum\limits_{k=N}^{n} 2^{-n}\frac {|x_n-y_n|} {1+|x_n-y_n|}<\sum\limits_{k=N}^{n} 2^{-n}<\epsilon$ if $N$ is sufficiently large.