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
$D$ 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 $\mathbb{R}$.