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this is my first question and I hope I don't make any relevant mistakes. For a little bit of context, in my real analysis homework I have the following problem.

Show that the subset D $\subset$ $l_2$ composed by all the sequences $(x_n)$ with $x_n\neq0$ being a rational number for a finite number of indexes, is countable and dense in $l_2$

Here the set $l_2$ is the set of all real number sequences $(x_n)$ that satisfy: $\sum_{n=1}^{\infty}|x_n|^2\leq \infty$ And the metric is given by:

$d((x_n)(y_n))=(\sum_{n=1}^{\infty}|x_n-y_n|^2)^\frac{1}{2}$

My question in reality is, should I consider D as being the set of all sequences with a finite number of entries being rational numbers that are different than 0 and the rest of the sequence being all real numbers. Or should i consider D as the set of all sequences with finite entries beign rational numbers that are different than 0 and the rest of all entries being equal to 0 ?

Not sure if the point of the question is clear, in any case hit me up and we can dicuss the matter. Thanks in advance, any help would be appreciated

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    $\begingroup$ All zero except for finitely many that are rational numbers. $\endgroup$ – user647486 Apr 18 at 2:09
  • $\begingroup$ Any suggestions as to how to show that this set is dense in l2 ? $\endgroup$ – Guilherme takata Apr 18 at 14:39
  • $\begingroup$ If $x\in \ell^2$, then there is $N$ such that $\sum_{k>N}|x_k|^2<\epsilon^2/2$. Then take $y\in\ell^2$ such that $y_k=0$ for $k>N$ and $y_k\in\mathbb{Q}$ and $\sum_{k=1}^{N}|y_k-x_k|^2<\epsilon^2/2$. It follows that $\|y-x\|<\epsilon$. $\endgroup$ – user647486 Apr 18 at 14:43
  • $\begingroup$ Oh, got it now. Thank you $\endgroup$ – Guilherme takata Apr 18 at 18:25
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$D$ is the set of all rational sequences with finitely many non-zero entries.

By a countability argument you can check that it's a countable set.

It's dense because you can truncate any sequence up to some term, the series that the norm must be convergent, and rationals are dense in $\mathbb{R}$.

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  • $\begingroup$ I get that sequences in D have finitely many non zero entries, but the rest of the sequence is entierly composed of zeros or by any real numbers ? $\endgroup$ – Guilherme takata Apr 18 at 14:38
  • $\begingroup$ If the (infinitely many) other entries were allowed to be any real number, then they could be non-zero. So those are the sequences that, after some entry, are all zeroes. $\endgroup$ – AspiringMathematician Apr 22 at 16:30

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