[Math] is the sum of the square of two real numbers greater than or equal to twice the product of the two real numbers

Question

I'm learning how to write proofs and cant seem to figure out how to do this one. Specifically Im interested if it is possible to prove the inequality by contradiction, contraposition, or by a direct proof; or if its possible at all.

Answer 1

$$x^2+y^2-2xy = x(x-y) + y(y-x) $$ $$= (x-y)(x-y)=(x-y)^2 \ge 0$$

Thus

$$x^2 +y^2 \ge 2xy,\, \forall x,y \in \mathbb R$$