how to prove this density result?

Question

Assume that $b\in \mathbb{C}$ such that $0<\vert b \vert <1$. We consider the familly $f_{p}=\{1,b^{p},b^{2p},b^{3p},b^{4p},...,b^{np},...)$. How can one prove that $\operatorname{Span}(f_{p}, \ p\in \mathbb{N}^{*})$ is dense in $l^{2}(\mathbb{C})$? Thanks.

Answer 1

Assume that $x$ is orthogonal to each $f_p$. The function $$h(z):=\sum_{k=0}^\infty x_kz^k$$ is holomorphic on the unit disk and vanishes on the non-discrete set $\{b^p,p\geqslant 1\}$. We conclude that $h$ vanishes identically hence $x=0$.