gustav1139
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Homework Statement
Suppose that \left\{a_{n}\right\} is a sequence of complex numbers with the property that \sum{a_{n}b_{n}} converges for
every complex sequence \left\{b_{n}\right\} such that \lim{b_{n}}=0. Prove that \sum{|a_{n}|}<\infty.
Homework Equations
The Attempt at a Solution
I tried going directly, at it, using the Cauchy condition on \sum{a_{n}b_{n}} to try to figure something out about the a_{n}, 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 \sum{|a_{n}|} diverges, then there exists a sequence \left\{b_{n}\right\} such that \sum{a_{n}b_{n}} 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 \lim{\frac{a_{n+1}b_{n+1}}{a_{n}b_{n}}}=r where r\in(0,1)), but that's unfortunately not a necessary condition for convergence.
So I'm stuck, and frustrated :(
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