If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

real-analysis sequences-and-series hilbert-spaces

Solution 1:

Start from here :$$(|a_n|-|b_n|)^2=a_n^2+b_n^2-2|a_nb_n|\ge 0$$

$$\implies |a_nb_n|\le \frac{1}{2}(a_n^2+b_n^2)$$By comparison test, $\sum a_nb_n$ is absolutely convergent , hence convergent.

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