Show that if $\displaystyle \sum\limits_{n=0}^\infty a_n^2$ and $\displaystyle \sum\limits_{n=0}^\infty b_n^2$ converge, then $\displaystyle \sum\limits_{n=0}^\infty a_nb_n$ converges.
Without using Cauchy Sharwz inequality?
If we for each $\displaystyle n$ indicate with $\displaystyle c_{n}^{2}$ the greater between $\displaystyle a_{n}^{2}$ and $\displaystyle b_{n}^{2}$, the series $\displaystyle \sum_{n=0}^{\infty} c_{n}^{2}$ is clearly convergent. But is $\displaystyle |a_{n}\cdot b_{n}|\le c_{n}^{2}$, so that the series $\displaystyle \sum_{n=0}^{\infty} a_{n}\cdot b_{n}$ is convergent...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$