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## Homework Statement

Suppose that [itex]\left\{a_{n}\right\}[/itex] is a sequence of complex numbers with the property that [itex]\sum{a_{n}b_{n}}[/itex] converges for

every complex sequence [itex]\left\{b_{n}\right\}[/itex] such that [itex]\lim{b_{n}}=0[/itex]. Prove that [itex]\sum{|a_{n}|}<\infty[/itex].

## Homework Equations

## The Attempt at a Solution

I tried going directly, at it, using the Cauchy condition on [itex]\sum{a_{n}b_{n}}[/itex] to try to figure something out about the [itex]a_{n}[/itex], but I got bogged down, and all the inequalities seemed to be pointing the wrong direction.

Then I tried to prove the contrapositive, that if [itex]\sum{|a_{n}|}[/itex] diverges, then there exists a sequence [itex]\left\{b_{n}\right\}[/itex] such that [itex]\sum{a_{n}b_{n}}[/itex] diverges as well. But I didn't get very far with that either.

I was able to prove it using the ratio test (i.e. if [itex]\lim{\frac{a_{n+1}b_{n+1}}{a_{n}b_{n}}}=r[/itex] where [itex]r\in(0,1)[/itex]), but that's unfortunately not a necessary condition for convergence.

So I'm stuck, and frustrated :(

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