Prove the Cauchy-Schwartz inequality: for a natural number \(n\) and real numbers \(a_1\), \(a_2\), ..., \(a_n\) and \(b_1\), \(b_2\), ..., \(b_n\) we have \[(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2).\]