(Cauchy–Schwarz inequality) Show that if $u$ and $v$ a two vectors, then $|\langle u, vangle|^2\le \langle u, uangle\cdot\langle v, vangle$. This inequality can also be written as $$|u_1v_1+u_2v_2+\cdots +u_nv_n|^2 \le (|u_1|^2+|u_2|^2+\cdots|u_n|^2)(|v_1|^2+|v_2|^2+\cdots|v_n|^2)$$