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?