Proof for existence of a particular sequence

Question

Suppose we have a sequence or real numbers $\{a_n\}_{n=1}^\infty$ such that $\sum |a_n|^2 \rightarrow\infty$ (diverges). How does one the prove the existence of another sequence $\{b_n\}_{n=1}^\infty$ such that $\sum a_nb_n\rightarrow\infty$ and $\sum |b_n|^2 <\infty$

Answer 1

Suppose that $\sum |a_nb_n|<\infty $ for all $b\in\ell^2$. This means that if we consider the family of linear functionals $\{f_k:\ell^2\to\mathbb C\} $, where $$f_k (b)=\sum_{n=1}^ka_nb_n. $$We have that, for each $b $, $$\sup_k|f_k (b)|\leq\sum|a_nb_n|<\infty. $$ By the Uniform Boundedness Principle, $$\sum|a_n|^2=\sup_k\|f_k\|^2 <\infty. $$

By the above, if $\sum|a_n|^2=\infty $, there has to exist some $b\in\ell^2 $ with $\sum |a_nb_n|=\infty $.