TITLE: Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$
QUESTION [6 upvotes]: Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$.

REPLY [2 votes]: Let $$y=x+a$$ $$a \neq 0$$ then $$2xy=2x(x+a)=2x^2+2ax<2x^2+2ax+a^2$$
see that the difference is exactly 'a' squared.