Given {$a_n$} ,{$b_n$} two monotone sequence of real numbers and that $\sum a_nb_n$ is convergent. Which one is the correct option ?
1) {$a_n$} is bounded and {$b_n$} is bounded
2) At least one of {$a_n$}, {$b_n$} is bounded .
My attempts: option is 1) that correct option take $a_n = b_n = \frac{1}{n^2}$,
option 2 is not correct because take $a_n = \frac{1}{n}$ and $b_n = n^2$,$\sum a_nb_n= \sum n$
Is my answer correct ?
On
I think the question is, if $\sum a_nb_n$ is bounded, which one must be true? In that case the answer is (2). It is easy to see that if they are both unbounded then (since they are monotone) so is $a_nb_n$, so the series does not converge.
An example to show that (1) is not always true is $a_n=n$ and $b_n=1/n^3$. In this case $a_n$ is unbounded, but $\sum a_nb_n=\sum 1/n^2$ converges.
Your attempt is wrong. You have showed that option 1 might be true and you did not show anything concerning option 2, because the example you gave does not satisfy the assumptions.
Option 1 is not correct in general, since for $a_n = n $ and $b_n = e^{-n}$ we have $\sum_n a_n b_n $ convergent and $a_n$ unbounded.
Option 2 is correct. Since $\sum_n a_n b_n$ converges, one has $|a_nb_n| \to 0$. If both were unbounded then we would not have $|a_nb_n|\to 0$. We are done.