How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?

Intuitive Approach
if $|x|\le |y|$ then we have
$$x\cdot y \le y\cdot y \le y.y + x\cdot x$$
$$\Rightarrow x\cdot y \le x^2+y^2\tag1$$
Using Symmetry $(1)$ holds if $|x|\ge |y|$ 
