This may be a silly question, but here goes. To ensure clarity, $\ell_1$ is the space of absolutely summable sequences, and $c_0$ the space of bounded sequences with limit 0. So we know that $\ell_1\subset c_0$ by basic principles. My question is: is $\ell_1$ when equipped with the sup-norm dense in $c_0$?
Here is my thought, and I would appreciate a comment on correctness or if something went wrong:
Let $\xi\in c_0$ and write $\xi=\{\xi_1,\xi_2,\xi_3,\dots\}$. Now define $P_n:\ell_1\to c_0$ by $P_n(\eta)=\{\eta_1,\eta_2,\dots,\eta_n\}$ So if $\xi\in c_0$, we can say $\xi=\underset{n\to\infty}{\lim}P_n\xi$
So does this get us all of $c_0$?
A typical example would be the harmonic sequence $\{1, 1/2, \dots, 1/n,\dots\}$. This is in $c_0$ but not $\ell_1$, but taking finite pieces of this sequence at a time guarantees us to remain in $\ell_1$, and we can approximate the sequence in $c_0$ as the limit of elements of $\ell_1$.