Show that if $\sum\limits_{n=0}^\infty a_n^2$ and $\sum\limits_{n=0}^\infty b_n^2$ converge, then $\sum\limits_{n=0}^\infty a_nb_n$ converges.
2. If we for each $n$ indicate with $c_{n}^{2}$ the greater between $a_{n}^{2}$ and $b_{n}^{2}$, the series $\sum_{n=0}^{\infty} c_{n}^{2}$ is clearly convergent. But is $|a_{n}\cdot b_{n}|\le c_{n}^{2}$, so that the series $\sum_{n=0}^{\infty} a_{n}\cdot b_{n}$ is convergent...
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