Convergence of $\sum_{i=1}^{\infty} a_ib_i$ using Cauchy-Schwarz inequality

Question

Let $a_n,b_n$ be complex numbers, and suppose that $\sum_{i=1}^{\infty}|a_i|^2$, $\sum_{i=1}^{\infty}|b_i|^2$ converge. Then, we have from Cauchy-Schwarz Inequality
$$
|\sum_{i=1}^{\infty} a_ib_i|^2\leq (\sum_{i=1}^{\infty}|a_i|^2)(\sum_{i=1}^{\infty}|b_i|^2),
$$
so $|\sum_{i=1}^{\infty} a_ib_i|$ converges. But, does it imply that $\sum_{i=1}^{\infty} a_ib_i$ converges? If I can prove that $\sum_{i=1}^{\infty} |a_ib_i|$ converges, then we are done. But I don't know how.

Answer 1

So that we can use Cauchy-Schwarz directly, let $A_i = |a_i|$ and $B_i = |b_i|$, then noting that $A_i, B_i > 0$,

$$ 0 \le \left(\sum_{i=1}^{\infty} |a_i b_i|\right)^2 = \left(\sum_{i=1}^{\infty} A_i B_i\right)^2 \le \sum_{i=1}^{\infty} A_i^2 \sum_{i=1}^{\infty} B_i^2 = \sum_{i=1}^{\infty} |a_i|^2 \sum_{i=1}^{\infty} |b_i|^2 < \infty. $$

The sequence $S_n = \sum_{i=1}^n |a_i b_i|$ is an increasing sequence of real numbers which is bounded above and thus converges. Hence $\sum_{i=1}^{\infty} |a_i b_i|$ exists and is finite and thus $\sum_{i=1}^{\infty} a_i b_i$ exists and is finite (absolute convergence implies conditional convergence).