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

Consider $s_n = \left(1, \frac12, \ldots, \frac1n, 0, 0, \ldots\right) \in \ell^2$. We have
$$\|s - s_n\|_2^2 = \left\|\left(0, , \ldots, 0, \frac1{n+1}, \frac1{n+2}, \ldots\right)\right\|_2^2 = \sum_{k=n+1}^\infty \frac1{k^2} \xrightarrow{n\to\infty} 0$$
because $\sum \frac1{k^2}$ is a convergent series. Therefore $(s_n)_{n=1}^\infty$ converges to $s$ in $\ell^2$.
