Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ also converge
then
Show if $\displaystyle\sum\limits_{k=1}^\infty {a_n}$ and $\displaystyle\sum\limits_{k=1}^\infty {b_n}$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_nb_n}$ also converge
then
Find two convergent series $\displaystyle\sum\limits_{k=1}^\infty {a_k}$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}$ such that $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ diverges.
My thought and attempt:
For the first part, By the theorem (If the series $\displaystyle\sum\limits_{k=1}^\infty {a_n}$ is convergent, then $\displaystyle\lim_{n\to\infty} a_n=0$),
if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, then $\displaystyle\lim_{n\to\infty} {a_k}^2=0$ and $\displaystyle\lim_{n\to\infty} {b_k}^2=0$ both equal to zero.
Then we can obtain both $(a_k)_{k=1}^\infty$ and $(b_k)_{k=1}^\infty$ converge
thus $\displaystyle\lim_{n\to\infty} {(a_k)^2\over|a_k|} = \displaystyle\lim_{n\to\infty} |a_k| = 0$, do the same thing for $b_k$
Since limit preserves arithmetic operation, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ converges.
Second part, I dont have any idea...
Third part is $a_n = b_n = {(-1)^n\over\sqrt{n+1}}$, am I right??
Thank you everyone... please help...