# Which of the following spaces are dense?

Let $X$ be the set of all real sequences, consider the subset $$S=\big\{x=\{x_n\}: x_n\in\mathbb{Q}~\forall n, x_n=0\text{ except for a finite number of n}\big\}.$$

We need to find out which of the followings are true.

• $a.$ $S$ is dense in $l_1$, the space of absolutely summable sequences, provided with the metric $$d_1(x,y)=\sum_{n=1}^{\infty}|x_n-y_n|\;.$$

• $b.$ $S$ is dense in $l_2$, the space of square summable sequences, provided with the metric $$d_2(x,y)=\left(\sum_{n=1}^{\infty}|x_n-y_n|^2\right)^{1/2}.$$

• $c.$ $S$ is dense in $l_{\infty}$, the space of bounded sequences, provided with the metric $$d_{\infty}(x,y)=\sup_n\{|x_n-y_n|\}\;.$$

My Intution: I know that $l_p$ is seperable for $1\le p<\infty$, as $S$ is countable and dense so $a,b$ are true, $c$ is false as $l_{\infty}$ is not seperable. Am I right?

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You’re right that $S$ is dense in $\ell_1$ and $\ell_2$, but can you actually prove it? It’s not enough to know that they are separable. – Brian M. Scott Jan 20 '13 at 11:53
No at the moment I can not prove it, I studied it many days before :(, is my guess for $c$ is also right? – La Belle Noiseuse Jan 20 '13 at 11:55
Here’s a hint for $\ell_1$: if $\epsilon>0$, there’s an $m$ such that $\sum_{n\ge m}|x_n|<\epsilon/2$, and you can choose rationals $q_0,\dots,q_{n-1}$ so that $\sum_{k=0}^{n-1}|x_k-q_k|<\epsilon/2$. – Brian M. Scott Jan 20 '13 at 11:58

Separability of $\ell_p$ by itself is not sufficient to state that some countable set is dense, but gives you a hint on how to construct an approximation sequence in $S$ for any sequences in $\ell_p$ when $p<\infty$. For the case $c$ consider the sequence consisting of $1$'s.
could you please tell me about the $c$, if I take $x_n=1\forall n$ then? – La Belle Noiseuse Jan 20 '13 at 12:33
@Panu: yes. Then the sup-norm of the difference with a vector that has at least one zero component (in your case it is in $S$ and thus has even more) is always $1$ – Ilya Jan 20 '13 at 12:35
Hence $c$ is not dense. – La Belle Noiseuse Jan 20 '13 at 12:37