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

\begin{align}
0\leq (x-y)^2 \implies
&
0\leq x^2-2xy +y^2 
\\
\implies
&
2xy\leq x^2+y^2
\\
\implies
&
xy\leq \dfrac{x^2+y^2}{2}
\\
\implies
&
xy\leq {x^2+y^2}
\end{align}
since the clearly nonnegative real $x^2+y^2$ clearly satisfies $\dfrac{x^2+y^2}{2} \leq x^2+y^2$.
