## Solution to Mathematics for Machine Learning Exercise 3.9

Let $n\in\mathbb N$ and let $x_1,\dots,x_n>0$ be $n$ positive real numbers so that $x_1+\dots+x_n=1$. Use the Cauchy-Schwarz inequality and show that a. $\sum_{i=1}^n x_i^2\geqslant \frac{1}{n}$. b. $\sum_{i=1}^n \frac{1}{x_i}\geqslant n^2$. Hint:…