Does $\sum a_i b_i $ converge, if $ \sum a_i^2 , \sum b_i^2 $ converge?

Question

I came across a question in linear algebra, Prove that $W = \{ \langle a_n \rangle \in V : \sum_{n=1}^\infty a_i^2$ is finite } is a vector space, where V is vector space of sequences in R. Now, I tried to proceed as usual, but got stuck in proving that $\sum_{n=1}^\infty {ab}$ is finite. This question is not from a standard textbook, and hence I am unsure whether the statement is even true.

With some examples, it is easy to see that this statement is definitely true for series of the form $\sum \frac{1}{n^k}, k\in R$. However, whether this extends to all sequences, I am not sure. If your answer is in the affirmitive, please provide a proof. I did try to prove the statement, but couldn't (Cauchy series doesn't help). I think Cauchy's inequality might help, but I am unsure how to use it(It has been time since I studied it in functional analysis class while in undergraduate class).

Answer 1

Just remembered the inequality in its correct form (credit- Wikipedia). The statement itself is a one-line proof: $|\langle u,v\rangle|^2 \leq \langle u,u\rangle\cdot\langle v,v\rangle$ I was writing it as $\sum{a_ib_i \leq (\sum a_i)^{1/2}(\sum b_i)^{1/2}}$ Credit : Qiaochu Yuan and Copper.hat in the comments section.