Originally Posted by

**tomcruisex** Show that if $\displaystyle \sum_{n=1}^{\infty}a_n^2$ and $\displaystyle \sum_{n=1}^{\infty}b_n^2$ are convergent then $\displaystyle \sum_{n=1}^{\infty}a_nb_n$ is absolutely convergent.

Hint: Show |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

I can prove the Hint -

$\displaystyle |xy|\leq 1/2(|x|^2 + |y|^2)$

$\displaystyle 2|xy| \leq |x|^2 + |y|^2$

$\displaystyle |x|^2 + |y|^2 - 2|xy| \geq 0$

$\displaystyle (|x| - |y|)^2 \geq 0 $

And this is true for all x, y as the square of any real number will be greater than or equal to 0.

But I cant see how to apply this to the original question?