I have the following sequence and limit:
$$\left(a_n\right)=\left(n-\sqrt{n}\left\lfloor{\sqrt{n}}\right\rfloor\right)$$ $$\lim_{n\to\infty}{n\left(\sqrt{n^2+2\ell}-n\right)}=\ell,$$
where $\ell$ is natural (i.e., $\ell \in \mathbb{N}$).
I have to prove that any natural number is a partial limit of the sequence $\left(a_n\right)$.
As I understand, I have to find a sequence of indexes that include $\ell$, that I can use to prove the claim. I've found a sequence of indexes that leads to a solution, but it is not a set of natural numbers. I can't get over this problem and build a sequence of natural indexes.
I thought of this 4 days, and I can't solve it, so I'm asking for help. This was my best solution, but the indexes are not natural numbers:
$$ \left(a_n\right)=\left(n-\sqrt{n}\left\lfloor{\sqrt{n}}\right\rfloor\right)= \sqrt{n}\left(\sqrt{n}-\left\lfloor{\sqrt{n}}\right\rfloor\right) $$ $$ \left(n_k\right)={\left(n+\frac {\ell}{n}\right)}^2$$ $$a_{n_k}=\sqrt{{\left(n+\frac {\ell}{n}\right)}^2}\left(\sqrt{{\left(n+\frac {\ell}{n}\right)}^2}-\left\lfloor{\sqrt{{\left(n+\frac {\ell}{n}\right)}^2}}\right\rfloor\right)=$$ $$=\left(n+\frac {\ell}{n}\right)\left(\sqrt{n^2+2\ell+\frac {\ell^2}{n^2}}-\lfloor{n+\frac {\ell}{n}}\rfloor\right)$$ $$\lim_{n\to\infty}{\left(n+\frac {\ell}{n}\right)\left(\sqrt{n^2+2\ell+\frac {l^2}{n^2}}-\lfloor{n+\frac {\ell}{n}}\rfloor\right)}=\lim_{n\to\infty}{n\left(\sqrt{n^2+2\ell}-n\right)}=\ell$$