Let $a_n$ be a sequence of real numbers.
Suppose that for any sequence of real numbers $(b_n)$, $$b_n \to 0 \implies \sum a_nb_n \;\; \text{converges}$$
Prove that $\sum|a_n|$ converges.
I attempted a proof by contradiction.
If $\sum|a_n|$ does not converge, then it diverges to $\infty$.
Therefore, $\displaystyle \frac{1}{\sum_{k=1}^n |a_k|}\to 0$ as $n$ goes to $\infty$.
As a result, $\displaystyle\sum_{n\geq 1} a_n\frac{1}{\sum_{k=1}^n |a_k|}$ converges.
How is that supposed to be a contradiction?
I must be on the wrong track...
