Hello again fmunshi!
Since bk is bounded, we know that there exists some bound M such that
|bk| < M for all k
We also know that sum(|ak|)=N, (where N, of course, is finite),
because sum(ak) is absolutely convergent.
We have to show that sum( |ak*bk| ) = L (where L, of course is a
finite number). This means that the series sum(ak*bk) converges
absolutely. Since M is a bound for we bk, we have
|ak*bk|
= |ak|*|bk|
<=|ak|*M
Therefore,
sum(|ak*bk|)
<=sum(M*|ak|)
=M*sum(|ak|)
=M*N
So, sum(|ak*bk|) is less than or equal to M*N (which is a finite
number) and greater than 0 (because |ak*bk| is greater than 0). This
implies that the series sum(|ak*bk|) converges to some number between
0 and M*N.
I hope this was clear enough. If you have any doubts regarding this
answer, please don't hesitate to request a clarification. Otherwise, I
await your rating and final comments.
Best wishes!
elmarto |