Cauchy-Schwarz inequality: $latex (a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2+b_2^2 + \cdots + b_n^2)$
When $latex a_1=\sqrt{x}, a_2=\sqrt{y}, b_1=\sqrt{y}, b_2=\sqrt{x}$, then $latex (2\sqrt{xy})^2\leq (x+y)^2$
Cauchy-Schwarz inequality: $latex (a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2+b_2^2 + \cdots + b_n^2)$
When $latex a_1=\sqrt{x}, a_2=\sqrt{y}, b_1=\sqrt{y}, b_2=\sqrt{x}$, then $latex (2\sqrt{xy})^2\leq (x+y)^2$