If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent.
Prove that if the positive term series $\sum^{\infty}_{n=1}a_n$ and $\sum^{\infty}_{n=1}b_n$ are convergent, also $\sum^{\infty}_{n=1}a_nb_n$ is convergent.
I've tried to solve it using comparison test, but no results.

One can also do these problems with Cauchy-Schwarz which states that for $a_n,b_n\geq 0,$ we have $\displaystyle \sum a_n b_n \leq \left(\sum a_n^2 \right)^{1/2} \left(\sum b_n^2 \right)^{1/2}.$ The second problem then follows by the argument in the comments of Brian's answer. It also gives $$\sum \sqrt{a_n a_{n+1} } \leq \left(\sum a_n \right)^{1/2}  \left(\sum a_{n+1} \right)^{1/2} $$ which solves the first problem.
