Show that $a^2b^2+c^2d^2 -2abcd \geq 0 $ with $a,b,c,d \in \mathbb{R}$

Question

Is it possible to Show that without argueing with Limits. I want to prove the Schwarz-Inequality with induction, if I could show this then I could prove the inductionstep.

Answer 1

Just use the fact that$$a^2b^2+c^2d^2-2abcd=(ab-cd)^2.$$