Real series/Cauchy-product/Absolute convergence/Fact

Let

$\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$

be absolutely convergent series of real numbers.

Then also the

Cauchy product ${}\sum _{k=0}^{\infty }c_{k}$ is absolutely convergent and for its sum the equation

${}\sum _{k=0}^{\infty }c_{k}={\left(\sum _{k=0}^{\infty }a_{k}\right)}\cdot {\left(\sum _{k=0}^{\infty }b_{k}\right)}\,$

holds.

Write a proof