In our notes there is a question :
"Show that $\overline {c_{00}}=\ell^{\infty}$ which means $c_{00}$ space is dense in $\ell^{\infty}$
but I have seen that sentence below
" $c_{00}$ is dense in $c_{0}$ and $\ell^1$ but not in $\ell^{\infty}$ "
$c_{00} =\{(x_n)_{n \in \mathbb N} : (x_n)=0 , \forall n\in \mathbb N, \exists N \in \mathbb N\}$
I've tried to show it is dense but I couldn't. My scribbles:
$\overline {c_{00}}=\ell^{\infty}$ iff $\forall x \in \ell^{\infty} ,\exists y_n \in c_{00}, y_n \to x$
-under $\|x\|_{\infty}=\sup|x_n|$ norm-
Let $x=x_n$ $\in \ell^{\infty}$ and $y_n \in c_{00}$
$x_n=x=(x_1,x_2,...,x_N,...)$ and $y_n=(y_1,y_2,...,y_N,0,0,...)$
$||y_n-x||_{\infty}=\|(y_1-x_1,y_2-x_2,...,y_N-x_N,-x_{N+1},...)\| _{\infty}=\sup\{|y_1-x_1|,|y_2-x_2|,...,|y_N-x_N|,|-x_{N+1}|,...\}$
Could you please about how should I continue and please correct me if there are mistakes..
Thanks