TITLE: Sum of squares finite for two sequences implies sum of products finite?
QUESTION [5 upvotes]: Let $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$ be sequences of real numbers such that $\sum_{i=-\infty}^\infty a_i^2$ and $\sum_{i=-\infty}^\infty b_i^2$ are both finite. Is it true that $\sum_{i=-\infty}^\infty |a_ib_i|$ is also finite?
This looks a lot like the Cauchy-Schwarz inequality, but the sum is an infinite sum, so I don't know how to apply it.

REPLY [4 votes]: Note that we have
$$\overbrace{\sum_{k=1}^n a_k b_k \leq \sqrt{\sum_{k=1}^n a_k^2} \sqrt{\sum_{k=1}^n b_k^2}}^{\text{Cauchy-Schwarz inequality for finite dimensional vector-space}} \leq \sqrt{\sum_{k=1}^{\infty} a_k^2} \sqrt{\sum_{k=1}^{\infty} b_k^2} = \text{constant}$$
Now let $n \to \infty$, to conclude what you want.