I want to show that $l_1$ is separable. I got the tip to
- Show that $C_{00}(\mathbb{N}) \subset l_1(\mathbb{N})$ (This I accomplished)
- Construct a sequence $(b_n)_n \in C_{00}(\mathbb{N})$ which approximates a given sequence $(a_n)_n \in l_1(\mathbb{N})$ arbitrarily well.
- Finally, construct a countable set which approximates $(b_n)_n$
My problem is within 2 and 3. I don't know what it means to approximate mathematically and how to do it. What are the constraints to call something a properly approximation?
I have looked through a lot of literature, but I couldn't find a section explaining how approximation really works. I don't need a fully fledged solution for the problem described above, more like the general idea how to approach such a task and how approximations work.
In part 2, given $\epsilon>0$ and a sequence $a_n \in \ell^1$, you want to find a sequence $b_n \in c_{00}$ with $\| b_n-a_n \|_{\ell^1}<\epsilon$. A convenient way to do this is to have $b_n$ just be a truncation of $a_n$. Figure out how to determine where to do the truncation.
I would say that part 3 is poorly worded. The issue is that $b_n$ is just fixed, so I can cheekily answer with $\{ b_n \}$ and technically be correct. But I can tell what they are trying to do because you said what the final goal is. The final goal is to find a fixed countable subset of $c_{00}$ (not depending on the choice of $b_n$) which contains a sequence converging to $b_n$. The natural choice of this is $c_{00} \cap \mathbb{Q}^\mathbb{N}$. You should show that $c_{00} \cap \mathbb{Q}^\mathbb{N}$ is dense in $c_{00}$, then by part 2 it must be dense in $\ell^1$ also.