# Useful inequalities

$\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}$ Cauchy-Schwarz inequality For $a_1,...,a_n,b_1,...,b_n\in\mathbb{R}$
$\left(\sum^n_{i=1}|a_i+b_i|^p\right)^\frac{1}{p}\le\left(\sum^n_{i=1}|a_i|^p\right)^\frac{1}{p}+\left(\sum^n_{i=1}|b_i|^p\right)^\frac{1}{p}$ Minkowski's inequality For $p\ge 1$ and $a_1,...,a_n,b_1,...,b_n\in\mathbb{R}$