The product of absconverg series and bounded seq is absolutely convergent

[icantadd]

Homework Statement

Assume $$\sum_{1}^{\infty} a_n$$ is absolutely convergent and {bn} is bounded. Prove $$\sum_{1}^{\infty} a_n * b_n$$ is absolutely convergent

Homework Equations

A series is absolutely convergent iff the sum of | an | is convergent

A series is convergent if for every e there is an N and P such that for all n >= N, for all p >0,
| S_{n+p} - S{n-1} | < e. S_k is the kth partial sum, and this basically says that the sequence of partial sum must be cauchy.

The Attempt at a Solution

because the series of an is abosultely convergent | |a_{n+p}| - |a_{n-1}| | < e
and bn is bounded means | b_n| < M

$$\left| \left| a_{n+p} b_{n+p} \right| - \left| a_{n-1}b_{n-1} \right| \right| \leq \left| a_{n+p} b_{n+p} \right| + \left| a_{n-1} b_{n-1} \right| \leq M \left| a_{n+p} \right| + M \left| a_{n-1} \right| < \frac{Me}{2M} + \frac{Me}{2M} = e$$

done

 

[rochfor1]

I'm confused...do you have a question?