Exercise 3.13

Since $\sum <!--\; FUNCTION\; APPLICATION\; -->|a_n|$ and $\sum <!--\; FUNCTION\; APPLICATION\; -->|b_n|$ converge, so does their Cauchy product $\sum <!--\; FUNCTION\; APPLICATION\; -->C_n$. Let $c_n$ be the Cauchy product of $a_n$ and $b_n$. Then $|c_n|\le C_n$, and as it is majorized by a convergent series, $\sum <!--\; FUNCTION\; APPLICATION\; -->|c_n|$ converges . I.e. $\sum <!--\; FUNCTION\; APPLICATION\; -->c_n$ converges absolutely.