The 1821 Cours d’analyse ends with nine notes, a kind of appendix with results Cauchy did not want to put in the main text. The second, on inequalities, contains two gems that every olympiad student knows. The inequality Cauchy’s theorem says that, for any real numbers $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, $$(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \le (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2),$$with equality only when the two lists are proportional.
So far, we have only worked with equalities, in equations we work with scales, which is the notion of balance. In this post we will work with the opposite, that is, imbalance. Difference between equality and inequality To illustrate the notion of equality and inequality, look at the following sentences: Letícia is 2 years older than Alexandre (equality); Letícia is at least 2 years older than Alexandre (inequality). In this case, we have different meanings. Let $L$ be Letícia and $A$ be Alexandre, we can write the sentences as: