Find a sequence converging to $\{1/k\}^{\infty}_{k=1}$ in $l^2$ Let $s=\{1/k\}^{\infty}_{k=1}$. Find a sequence $\{s_n\}^{\infty}_{n=1}$ of points in $l^2$ such that each $s_n$ is distinct from $s$ and such that $\{s_n\}^{\infty}_{n=1}$ converges to $s$ in $l^2$.
This is a problem from Goldberg. Exercise 4.3, 3. I don't understand the question. How can I find a sequence that converges to another sequence? Is $s$ the limit of $\{1/k\}$ or it is the variable assigned to the sequence itself? If it is the limit then it can be worked out

Hint. Recall that $l^2$ is the normed space of all square summable sequences $s$, such that
$$\|s\|^2_2=\sum_{k=1}^{\infty}s(k)^2<+\infty.$$
Let $s_n(k):=\frac{1}{k}+\frac{1}{nk}$ (also $s_n(k):=\frac{a_n}{k}$ with $a_n\to 1$  will work), then  $\{s_n(k)\}_{k\geq1}\in l^2$ for each $n\geq 1$. Now consider the limit of
$$\lim_{n\to \infty}\|s_n-s\|^2_2=\lim_{n\to \infty}\sum_{k=1}^{\infty}\left(s_n(k)-\frac{1}{k}\right)^2$$
and show that it is zero.
