(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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**