1. ## Absolute Convergence Question

I tried using the definition of absolute convergence (similar to Cauchy Criterion), but had trouble reaching at ajbj.
Any help would be greatly appreciated.
Thanks

2. apply the Cauchy-Schwarz inequality.

3. I understand

but how do I get rid of the square signs on sequence a and b?
I can't make a connection

4. Here's what I just tried.

but the absolute sign is around the whole summation, not just the term.
Am I doing this right?

5. i wasn't actually talking about that kind of application, what i meant is $\sum\limits_{j=1}^{\infty }{a_{j}b_{j}}\le \sqrt{\sum\limits_{j=1}^{\infty }{a_{j}^{2}}\cdot \sum\limits_{j=1}^{\infty }{b_{j}^{2}}}.$

(ohhh, may be it's the same, it's just that i'm not so familiar with norms.)