It's been asked to prove that the following set of vectors in $\ell^2(\mathbb{N})$ is total (its span is dense in $\ell^2(\mathbb{N})$):
$$ \{c^n := (1,n^{-1},n^{-2}, \ldots) \}_{n=2}^\infty. $$
And the hint for that is to use the fact that for any $a \in \ell^2(\mathbb{N})$, the function $f(z) := \sum_{j \in \mathbb{N}}a_jz^{j}$ is holomorphic on the unit disc.
This question has been asked on the part in which hilbert spaces and orthonormal bases are discussed. I don't know how one can relate holomorphy to this. In fact function $f(z)$ is equal to the inner product $\langle a,c^n \rangle$ if we put $z = n^{-1}$. But from here onwards I have absolutely no clue how to proceed.
As requested, here is my comment promoted to an answer:
A further hint: it suffices to prove if $\langle a, c^n \rangle = 0$ for all $n$, then $a = 0$. And a nonzero holomorphic function has all its zeros isolated, so in particular if a holomorphic function has a sequence of zeros converging to some point in its domain, then that holomorphic function must be identically zero.