convergent and bounded sequence

Question

Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| ≤ 1$ for every positive integer $n$, $\ell^1$ be the set of all real sequences $\{a_n\}$ such that $\sum a_n$ converges absolutely, $\ell^\infty$ be the set of all real sequences $\{a_n\}$ such that $|a_n|$ is bounded, and $c_0$ be the set of all real sequences $\{a_n\}$ such that $\lim_{n \rightarrow} a_n = 0$.

Let $\{a_n\}$ be a sequence such that $\{a_n b_n\} \in \ell^1$ for every sequence $\{b_n\} \in \ell^1$. Prove that $\{a_n\} \in \ell^\infty$. This statement, is it false if $\ell^\infty$ is replaced by $c_0$?

Answer 1

Suppose $\{a_n\}$ is unbounded. Then there is a subsequence $\{a_{n_j}\}$ with $a_{n_j} > j^2$. Taking $b_n = 1/j^2$, we then see $\sum a_n b_n$ does not converge. This statement is false for $c_0$ since we may take $a_n = 1$ constant.