(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is part of a question on absolute convergence on series. The following equation is given as a hint. It says that before answering the question on series I should prove that |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

2. Relevant equations

3. The attempt at a solution

I know that this is the same as |x||y| <= 1/2(|x||x| + |y||y|). That is about all I am able to do to manipulate this equation. I tried solving for x or y but found it is not possible to separate them out. So I can't see how to prove this other than -

letting x = 0, then showing the equation is true for a few values of y

letting x = 1, then showing the equation is true for a few values of y

etc...

And then saying the equation is true 'by induction'. Surely there is a better, less awkward, way to show that |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Show that this inequality is true for all x, y ε R

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