I am having a hard time proving this result. Could someone give me a hand? Thanks.

If $\displaystyle \sum {a_n}^2$ and $\displaystyle \sum {b_n}^2$ converge, then $\displaystyle \sum a_n b_n$ converges absolutely.

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- Dec 8th 2012, 03:54 PMjackieConvergence of series
I am having a hard time proving this result. Could someone give me a hand? Thanks.

If $\displaystyle \sum {a_n}^2$ and $\displaystyle \sum {b_n}^2$ converge, then $\displaystyle \sum a_n b_n$ converges absolutely. - Dec 8th 2012, 04:11 PMGJARe: Convergence of series
Hi jackie,

I think we can get what you're looking for using the Cauchy-Schwarz inequality. Give it a shot with this and let me know how it goes. Good luck! - Dec 9th 2012, 08:53 PMjackieRe: Convergence of series
Thank you for your help GJA. By using the Cauchy-Schwarz inequality I am able to get a bound for the sequence of partial sums of the series $\displaystyle \sum \mid a_n b_n \mid$. Clearly, the sequence of partial sums is increasing. That should be enough to show convergence right?