Show that for any reals $a; b; x; y \ge 0, (ax + by)^2 \le (a^2 + b^2)(x^2 + y^2)$. (Hint: Use Cauchy-Schwarz inequality in $\mathbb{R}^{2}$.)
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.